Problem: Find $(\log_2 x)^2$ if $\log_2 (\log_8 x) = \log_8 (\log_2 x)$. Solution: I honestly took a while doing this one because I didn't see we could write $\log_8x = \frac{\log_2x}{\log_{2}8}$ which solves the problem very quickly. Here was my approach: We can see that $\log_2(\log_8 x) = \log_8 ( (\log_8 x)^3)$. Subtracting $\log_8 (\log_2 x)$ from both sides, we get that $ \log_{8}( \frac{(\log_8x)^3}{\log_2x})=0 \implies \frac{(\log_8x)^3}{\log_2x} = 1 \implies (\log_8x)^3 = \log_2x .$ From here, we can raise each side of $8=8$ to each side of our equality above to get that $x^{(\log_8x)^2} = x^3 \implies \log_8x = \sqrt{3} \implies x=2^{3\sqrt{3} }$ so the answer is $27$.