exponential equations Archives • Jill Hacker

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:

$5^{3x}=6$

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:

$a^b=c<=>\log_a c=b$

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 $\log_a c=b$ gives:

$\log_5 6=3x$

Divide both sides by 3 and switch the sides:

$x=\frac{\log_5 6}{3}$

That’s the answer.

Caution

Don’t try to cancel the 6 in the numerator with the 3 in the denominator. $x=\log_{5}6$ 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.

$5^{3x}=6$	This is the equation that you have to solve for x.
$\log_{5}5^{3x}=\log_{5}6$	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.
$3x\log _5 5= \log_5 6$	The power rule for logarithms says $\log a^b=b\log a$ . In other words, you can bring that exponent down. Apply this rule to the equation’s left-hand side: $\log _5 5^{3x}=3x\log _5 5$
$3x \cdot 1=\log_5 6$	$\log _a a= 1$ so $\log _5 5= 1$ . Substitute 1 for $\log _5 5$ .
$\fn_jvn {\color{Blue}x=\frac{\log _{5}6}{3} }$	Divide both sides by 3 and there’s the answer.