# How to solve exponential equations

You will learn how to solve exponential equations by taking the common logarithm of each side of an exponential equation.

## Examples showing how to solve exponential equations

Example #1

23x = 64

Take the common logarithm of each side.

log 23x = log 64

Use the power property of logarithms

3x log 2 = log 64

Divide each side by 3 log 2 to isolate x

x = log 64 / 3 log 2

x = 1.8 / 3 x 0.3   ( log 64 and log 2 were rounded to the nearest 10th)

x = 1.8 / 0.9

x = 2

Notice that you could have solved the problem above this way too

23x = 64

23x = 26

3x = 6

x = 2

However, this works only if you can easily rewrite the number on the right or 64 so that it will have the same base or 2 as the expression on the left. For the next example, it is not possible to easily rewrite the expression so that the same base shows up. Can you tell what y is for this expression 2y = 60 ?

Therefore, it is easier to take the log of both sides.

Example #2

23x = 60

Take the common logarithm of each side.

log 23x = log 60

Use the power property of logarithms

3x log 2 = log 60

Divide each side by 3 log 2 to isolate x

x = log 60 / 3 log 2

x = 1.78 / 3 x 0.3   ( log 60 and log 2 were rounded to the nearest 10th)

x = 1.78 / 0.9

x = 1.97

Example #3

52x+4 = 120

Take the common logarithm of each side.

log 52x+4 = log 120

Use the power property of logarithms

(2x + 4) log 5 = log 120

2x log 5 + 4 log 5 = log 120

2x log 5 = log 120 - 4 log 5

Divide each side by 2 log 5 to isolate x

x = ( log 120 - 4 log 5 ) / 2 log 5

x = ( 2.079 - 4 x 0.699 ) / 2 x 0.699   ( log 120 and log 5 were rounded to the nearest thousandth)

x = ( 2.079 - 2.796 ) / 1.398

x = -0.717 / 1.398

x = -0.512

## How to solve exponential equations when there are two exponential expressions

Example #4

Solve 34x + 3 - 8-x+2 = 0

Step 1: Isolate the exponential expression

34x + 3  = 8-x+2

Step 2: Take the common logarithm of each side

log 34x + 3  =  log 8-x+2

Step 3: Use the power property of logarithms

(4x + 3) log 3 = (-x + 2) log 8

Step 4: Solve for the variable

4x log 3 + 3 log 3 = -x log 8  + 2 log 8

4x log 3 + x log 8  = 2 log 8 - 3 log 3

x(4 log 3 + log 8) = 2 log 8 - 3 log 3

x = (2 log 8 - 3 log 3) / (4 log 3 + log 8)

x = (2 × 0.90 - 3 × 0.47) / (4 × 0.47 + 0.90)

x = 0.39 / 2.79 = 0.139

Generally speaking, follow these four steps when solving exponential equations.

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