Solving Exponential Solving Exponential and Logarithmic and Logarithmic

EquationsEquations

Solving Exponential Solving Exponential and Logarithmic and Logarithmic

EquationsEquations

Exponential Equations are equations of the form y = abx. When solving, we

might be looking for the x-value, the b value or the

y- value. First, we’ll review algebraic methods.

354 2b

When solving for b, isolate the b value; then raise both sides of the equation to the reciprocal power of the exponent.

3

113 33

27

27

3

b

b

b

When solving for y, solve by performing the indicated operations.

1

216y

1

21

16

1

4

y

y

When solving for the exponent, rewrite the bases so they have the same base. If the bases are equal,

the exponents are equal. Now, solve.18 4x x

2 132 2

3 2( 1)

3 2 2

2

x x

x x

x x

x

Both bases now equal 2 so we can just use the equal exponents.

Solve the following exponential equations:

3

2

3

1

81 3

125

19

3

xx

b

y

3 = b

y = 25

Solving Exponential Equations by Using the Graphing Calculator

• Always isolate the variable FIRST!! • Graph the function in Y1

• Graph the rest of the equation in Y2

• Use the intersect function (found with ) to determine the

x value

Solve for x to the nearest thousandth: ex=72

• Graph Y1 = ex and Y2= 72

• Use intersect to give the answer

x =4.277

Log Equations are of the form

y=logba or

y= ln x where the base is e.

To solve for x, we need to undo the log format by rewriting in exponential form.

Y= log216 4=log3x 3= logx1000

2y=16 34=x x3=1000

Now we use the exponential rules to solve.

2y=16 81 = x X3 = 103 y=3

x=4

Solve algebraically: 72xe ln ln 72xe

ln 72

4.277

x

x

Rewrite as an ln equation:Since ln ex means the exponent of ln ex

, just use x:

Note: when an equation is written in terms of e, you MUST use natural logs. Otherwise you may use log or ln at will.

Solve for x algebraically: 2x = 14

Take the ln or log of each side and solve.

ln 2 ln14

ln 2 ln14

ln14

ln 23.807

x

x

x

x

log 2 log14

log 2 log14

log14

log 2

3.807

x

x

x

x

Solve for x: 2x=14 graphically.

• Graph Y1 = 2 x and Y2= 14

• Use intersect to approximate the answer

X=3.807

3

ln 3x

e x

Solve lnx = 3 to the nearest tenth.

X = 20.1

Solve for x:

2 32x

19

3

x

X= 5

X=-2

More Involved Equations

Sometimes log or ln equations require a few more steps to “clean them up” before we can simply “take the log” or “undo the log”.

5 2ln 4x Before you can take the ln, you need to isolate it!

2ln x = -1

ln x = -0.5

e-0.5=x

x= 0.60653…

To solve graphically, you still must isolate the ln expression.

2ln x = -1

Graph as y1 = 2lnx

y2= -1

Solve ln 3 2x

2e 3 definition of log

2.46

x

x

5Solve 2 log 3 4x

5

2

log 3 2

5 3 log definition

258.3333

3

x

x

x

2

Solve using the graphing calculator:

ln 2x x 2ln 2 0x x

More practice

3(2 ) 42x

38 360x

log 2 log14

log 2 log14

log143.807354922

log 2

x

x

x

3

3

8 360

ln8 ln 360

3 ln8 ln 360

ln 3603

ln8ln 360

3ln8.93453023

x

x

x

x

x

x

Solving Logarithmic Equations Algebraically Using Laws of

Logarithms

When an equation contains the word log or ln, we need to eliminate it to solve the equation so first we apply the laws of logarithms to “undo” the addition by changing to multiplication, “undo” subtraction by changing it to division, and “undo” powers by changing them to multiplication..

Solve: Log 2(4x+10) – log2(x+1) = 3

Log 2(4x+10) – log2(x+1) = 3

24 10

log 31

x

x

34 102

1

x

x

Apply Quotient Rule.

Definition of Logarithm

4 10 8( 1)

4 10 8 8

2 4

1

2

x x

x x

x

x

Cross multiply and solve

2Solve 4 3 2xe

2

2

2

4 5

5

45

ln ln4

52 ln

41 5

ln 0.112 4

x

x

x

e

e

e

x

x

Solve: 3 15 7x x

2 3 2 0x xSolve e e

2 5Solve 2(3 ) 4 11t 2 5

2 5

2 5

2 53 3

3

3

2(3 ) 4 11

2(3 ) 15

153

215

log (3 ) log2

152 5 log Inverse Property

25 15

t= log 3.422 2

t

t

t

t

t