How to solve exponential equations
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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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