Problem:
Suppose that a,b, and c are positive real numbers such that alog37=27, blog711=49, and clog1125=11. Find
a(log37)2+b(log711)2+c(log1125)2
Solution:
It follows from the properties of exponents that
a(log37)2+b(log711)2+c(log1125)2=(alog37)log37+(blog711)log711+(clog1125)log1125=27log37+49log711+11log1125=33log37+72log711+1121⋅log1125=73+112+25=343+121+5=469.
The problems on this page are the property of the MAA's American Mathematics Competitions