4.4 Solving Exponential and Logarithmic

Equations

Solving Exponential Equations:

If possible, express both sides as powers of the same base

Equate the exponents

Solve for variable

3 1 2 2 7

4 4 14

3 3

3 3

( )x x

x x

x x 4 4 14

1 2 727 3 9( )x x

4 3 14

18 3

6

x

x

x

3 1 2 2 73 3 3( ) ( )x x

Solving Exponential Equations

If it’s not possible to express both sides as powers of the same base: Isolate the exponential expression Take the log of both sides Use the rules for logs to “break down” the

expressions Solve for the variable

Solving Exponential Equations

Solve: Take the log of each side Use the rules for logs to

“break down” the expression

Solve for the variable

(check your answer!)

3 15 7log ( ) log ( )x xb b

3 15 7x x

Any base can be used, and since you’ll want to use your calculator, that will probably

be 10

x 0.675

3 5 1 7

3 5 7 7

log ( ) log

log log logb b

b b b

x x

x x

3 5 7 7

3 5 7 7

log log log

( log log ) logb b b

b b b

x x

x

7

3 5 7

log

log logb

b b

x

Solving Logarithmic Equations:

Use the rules for logs to simplify each side of the equation until it is a single log or a constant:

log log log ( )

log log log ( )

log ( ) log ( )

log ( ) log ( )

2 2 2

2 22

22

22

22

2 22

2 5 2 6

5 6

5 6

25 6

x x

x x

x x

x x

log log log

log log log ( )

log log

log

5 5 5

5 52

53

5 5

5

2 7 125

7 5

49 3

493

x

x

x

x

Solving Logarithmic Equations

Log = Log

Equate the arguments

Solve the resulting equation

Reject solutions that would mean taking the log of a negative number!

225 6( )x x

22 225 6log ( ) log ( )x x

2

2

25 12 36

0 37 36

0 36 1

36 0 1 0

36 1

( )( )

x x x

x x

x x

x x

x x

Solving Logarithmic Equations

Log = Constant

Turn logarithm into an exponential

Solve and check

5 349

logx

3549

x

12549

125 49 6125

x

x