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: Use 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? 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.