1988 AIME Problems/Problem 3

Problem

Raise both as exponents with base 8:

$\begin{eqnarray*}8^{\log_2 (\log_8 x)} &=& 8^{\log_8 (\log_2 x)}\\2^{3 \log_2(\log_8x)}} &=& \log_2x\\(\log_8...$

A quick explanation of the steps: On the 1st step, we use the property of logarithms that $a^{\log_a x} = x$ . On the 2nd step, we use the fact that $k \log_a x = \log_a x^k$ . On the 3rd step, we use the change of base formula, which states $\log_a b = \frac{\log_k b}{\log_k a}$ for arbitrary $k$ .