Problem:
Determine the value of ab if log8​a+log4​b2=5 and log8​b+log4​a2=7.
Solution:
Adding the two equations and using standard 10 properties yields
log8​a+log8​b+log4​a2+log4​b2=log8​(ab)+2log4​(ab)=12
Moreover, since log8​x=(log2​8)(log2​x)​=31​log2​x, and similar1y, log4​x=(log2​4)(log2​x)​=21​log2​x, the above equation is equivalent to
34​log2​(ab)=12
It follows that log2​(ab)=9, and hence ab=29=512​.
The problems on this page are the property of the MAA's American Mathematics Competitions