4-6 Solving Exponential and Logarithmic Equations

9 9 2 2

x

2 9

x

2

-6x

x x

Ex 1: Solve the equation.

    6 81 3  

-6x

x

9  6 3

x x

4

x

   4 2 2

One possible way to solve exponential equations is to find the same base on each side of the equal sign.

 4

x

  4

-4

x

 1

-4

Ex 2: Solve the equation

5

x

 8

2 nd way: rewrite as a log and use the change of base formula.

log 5 log 8 8  log 5 

x x

.

903  .

699 1 .

29 

x x

Ex 3: Solve

8  10 5

x

 4  35

-8

10 5

x

 4 log 10 5

x

 4 

-8

27  log 27 5

x

 4  log 27

Goal: Isolate the constant with the variable exponent.

Use logarithms to eliminate the base. (Note: make sure to use the same log on both sides of the equal sign).

Ex 3 cont…

5

x

 4  log 27

-4 -4

5

x

5

x

  log log 27

5

27   4 4 5

x x

 1 .

43  4  5  .

51

Ex 4: Solve

log 4 (

x x

 3 )  3   log 8

x

4  ( 8 17

x

-3 -3

 17 )

If you have a logarithm with the same base, you can set them equal to each other.

x

 8

x

-8x -8x

 14  7

-7

x

 14

-7

x

  2

ln 

x

 4   2

+4

e

2 

x



+4

4

e

2  4 

x

Ex 5: Solve One log on one side of the equal sign = rewrite in exponential form.

11 .

39 

x

Either simplified or rounded is a correct answer.

Ex 6: Solve the equation. Check for extraneous solutions.

Ex 6: Solve the equation. Check for extraneous solutions.