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.

How to solve exponential equations

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