Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1

Name _____________________________ Class ___________________Date ____________

Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4

10log3 log 3x x+ +=

e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =

Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…

Changing loge x y= to exponential form would give ye x= .

Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.

1. 5e ________________ 2. 4e− ________________ 3. 1

3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.

4. 4 ln 2 ln f− = ________________ 5. 1

lnx 3ln2

y+ = ________________

Solving natural logarithmic equations.

Solve ( )2ln 3 5 4x + =

Write in exponential form. ( )24 3 5e x= +

Take the square root of both sides. 4 3 5e x± = +

Subtract 5 from both sides. 4 5 3e x± − =

Divide both sides by 3. 4 5

3

ex

± − =

Evaluate using the calculator. 7.39 and 4.130x x= − =

6. Solve ln 9 5x = 7. Solve 2

ln 123

x + =

Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=

Divide by 2. ln 2.5

x2

=

Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =

Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =

13. ( )ln 4 1 36x − =

14. 5 4 7x

e + = 15. 22ln 2 x 1=

lne = 1