AIME 1988 P3

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Problem Statement

Find $(\log_2 x)^2$ if $\log_2 (\log_8 x) = \log_8 (\log_2 x)$.

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Solution

$$\log_2 (\log_8 x) = \log_8 (\log_2 x)$$ $$\log_2 (\dfrac{\log_2(x)}{\log_2(8)}) = \dfrac{\log_2(\log_2(x))}{\log_2(8)}$$ $$\log_2 (\dfrac{\log_2(x)}{3}) = \dfrac{\log_2(\log_2(x))}{3}$$ $$\log_2 (\log_2(x)) - \log_2 (3) = \dfrac{\log_2(\log_2(x))}{3}$$ $$3\log_2 (\log_2(x)) - 3\log_2 (3) = \log_2(\log_2(x))$$ $$2\log_2 (\log_2(x)) = 3\log_2 (3)$$ $$\log_2 ((\log_2(x))^2) = \log_2 (3^3)$$ $$\therefore (\log_2 x)^2 = 27$$