8.5 – Exponential and Logarithmic Equations. CHANGE OF BASE FORMULA where M, b, and c are positive...
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Transcript of 8.5 – Exponential and Logarithmic Equations. CHANGE OF BASE FORMULA where M, b, and c are positive...
8.5 – Exponential and Logarithmic Equations
CHANGE OF BASE FORMULA
6826.15log
15log15log5
where M, b, and c are positive numbers and b, c do not equal one.
b
MM
c
cb log
loglog
Ex: Rewrite log515 using the change of base formula
Steps for solving exponential equations
Take a common logarithm of each side
Use the power property of logarithms
Solve for x by dividing Use a calculator to find the
approximate value
Solving Exponential Equations
43 x
43 x
4log3log x
3log
4logx
4log3log x 1. Take the log of both sides
2. Use the power property
3. Solve for x.
Solve . Round to the nearest ten-thousandth.
X=1.2619 4. Use a calculator.
Check your answer – 31.2619=4
Another Example
1013 4 x
1013 4 x
101log3log)4( x
3log
101log4 x
101log3log 4 x 1. Take the log of both sides
2. Use the power property
3. Solve for x.
Solve . Round to the nearest ten-thousandth.
X=4.2009 – 4 = 0.2009 4. Use a calculator.
Check your answer – 30.2009+4=101
Let’s try some
1505x 802 x4
Let’s try some
1505x 802 x4
Let’s try some
735 x 2073 x 1003 4 x
Let’s try some
735 x 2073 x 1003 4 x
CHANGE OF BASE – HOW IT WORKS
Use the change of base formula to evaluate . Then convert it to a logarithm of base 2.
15log3
4650.215log3
x23 log15log
3log
15log15log3 1. Rewrite using the
change of base formula
2. Use a calculator
3. Write an equation to convert to base 2
CHANGE OF BASE – HOW IT WORKS
x2log4650.2
2log
log4650.2
x
xlog7420.0
7420.010x
xlog2log 4650.2 6. Multiply both sides of the equation by log2
7. Use a calculator; simplify.
8. Write in exponential form.
5. Rewrite using the change of base formula
4. Substitute log315=2.4650
X=5.5208 9. Use a calculator.
Log315 is approximately equal to 2.4650 or log25.5208
Let’s try one
Use the change of base formula to evaluate . Then convert it to a logarithm of base 8.
400log5
7227.3400log5
x85 log400log
5log
400log400log5 1. Rewrite using the
change of base formula
2. Use a calculator
3. Write an equation to convert to base 2
x8log7227.3
8log
log7227.3
x
xlog3619.3
3619.310x
xlog8log .72273 6. Multiply both sides of the equation by log8
7. Use a calculator; simplify.
8. Write in exponential form.
5. Rewrite using the change of base formula
4. Substitute log5400=3.727
X=2301 9. Use a calculator.
Log5400 is approximately equal to 3.7227 or log82301
SOLVING SIMPLE LOG EQUATIONS
x642
3
8
6
16x
26log
232log
4
4
x
x
2)3(log2log solve tologs of properties Use 44 x
1. Use the product property
2. Write in exponential form.
x616
2)3(logx2log 44
3. Simplify
4. Solve for x.
Let’s try some
1643 555 loglogxlog
Let’s try some
1643 555 loglogxlog
Let’s try some
64log4logxlog2
Let’s try some
64log4logxlog2
Solving exponential equations with a graphing calculator
150062 x
1. Type two equations into y=
Solution: 2.0408
2. Graph. Suggest Zoom fit (0)especially for large values
3. Use the calc function to find the intersection of the two graphs.