Solve 9 to the power x, divided by 12, equals 21.

Step-by-step solution

1. 1

   Isolate the exponential before touching logarithms. Multiply both sides by $12$:

   $9^x=21\cdot 12=252$

   Taking logarithms first, while the $12$ is still attached, would drag an extra $\log 12$ through every line for no benefit.

2. 2

   Check whether 252 is a power of 9. $9^2=81$ and $9^3=729$, and $252$ lies strictly between them. So the answer is between $2$ and $3$ and cannot be a nice fraction — a logarithm is genuinely required.

3. 3

   Take a logarithm of both sides. Applying $\log_9$ directly, or any base with the change-of-base formula:

   $x=\log_9 252=\frac{\ln 252}{\ln 9}$

   The two logarithms must be in the same base; natural logs are the usual choice.

4. 4

   Evaluate. $\ln 252=5.529429$ and $\ln 9=2.197225$, so

   $x=\frac{5.529429}{2.197225}=2.5165516$

   This sits between $2$ and $3$, as step 2 predicted.

5. 5

   Verify by substitution. $9^{2.5165516}=252.00$, and dividing by $12$ returns $21.00$. So $x=\log_9 252\approx 2.5166$. (An equivalent exact form is $x=\frac{\ln 252}{2\ln 3}$, since $\ln 9=2\ln 3$.)