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How to Solve an Exponential Equation

This post about how to solve an exponential equation is part of a series of posts to help you prepare for the Advanced Algebra and Functions part of the Accuplacer test.

Question

Solve for x:


Solution

You are asked to solve this exponential equation.

How do you get that x out of the exponent so you can solve for it? Consider the relationship between exponents and logarithms:

where “<=>” means the equations on both sides of the symbol are equivalent; each is true only when the other is true.

Note that the equation on the right-hand side is solved for b, the exponent. Let’s apply that to the equation we’re given. 5 stands in the place of a, 3x in the place of b, and 6 in the place of c. Substituting those values into gives:

Divide both sides by 3 and switch the sides:

That’s the answer.

Caution

Don’t try to cancel the 6 in the numerator with the 3 in the denominator. is a function carried out on the number 6; it is not a multiple of 6, so it does not cancel with 3.

Another Way to Solve an Exponential Equation with Logs

Take the log of each side.

This is the equation that you have to solve for x.
If two things are equal, their logs are equal. Why log base 5? Any base is valid; choosing the base that’s in the problem will likely give you the cleanest answer.
The power rule for logarithms says . In other words, you can bring that exponent down. Apply this rule to the equation’s left-hand side:
so . Substitute 1 for .
Divide both sides by 3 and there’s the answer.

Related: “Exponential Growth.”

This question is similar to question number 18 in the sample questions for the Accuplacer Advanced Algebra and Functions test.