Problem:
What is the value of
log402log280−log202log2160?
Answer Choices:
A. 0
B. 1
C. 45
D. 2
E. log25
Solution:
Answer (D): Recall that logab is the reciprocal of logba. Therefore the given expression equals
log280⋅log240−log2160⋅log220=log2(24⋅5)⋅log2(23⋅5)−log2(25⋅5)⋅log2(22⋅5)
=(4+log25)(3+log25)−(5+log25)(2+log25)=12+7log25+(log25)2−10−7log25−(log25)2=2.
The problems on this page are the property of the MAA's American Mathematics Competitions