Problem:
Find (log2​x)2 if log2​(log8​x)=log8​(log2​x).
Solution:
Since log8​x=logx​81​=3logx​21​=31​log2​x and log8​(log2​x)=31​log2​(log2​x), the given equation is equivalent to
log2​(3y​)=(31​)log2​y(1)
where y=log2​x. From (1), log2​(3y​)3=log2​y, hence (3y​)3=y; i.e.,
y(y2−27)=0(2)
Since yî€ =0 (for otherwise, neither side of (1) would be defined), it follows from (2) that y2=(log2​x)2=27​.
The problems on this page are the property of the MAA's American Mathematics Competitions