Problem:
Suppose that a,b, and c are positive real numbers such that alog3​7=27, blog7​11=49, and clog11​25=11​. Find
a(log3​7)2+b(log7​11)2+c(log11​25)2
Solution:
It follows from the properties of exponents that
​a(log3​7)2+b(log7​11)2+c(log11​25)2=(alog3​7)log3​7+(blog7​11)log7​11+(clog11​25)log11​25=27log3​7+49log7​11+11​log11​25=33log3​7+72log7​11+1121​⋅log11​25=73+112+25​=343+121+5=469​.​
The problems on this page are the property of the MAA's American Mathematics Competitions