Problem:
What is the value of
(log5)3+(log20)3+(log8)(log0.25)
where log denotes the base-ten logarithm?
Answer Choices:
A. 23
B. 47
C. 2
D. 49
E. 3
Solution:
The given expression equals
(log210)3+(log10⋅2)3+(log23)⋅log2−2=(1−log2)3+(1+log2)3+(3log2)(−2log2)
Expanding the cubes and simplifying yields
1−3log2+3(log2)2−(log2)3+1+3log2+3(log2)2+(log2)3−6(log2)2=(C)2
The problems on this page are the property of the MAA's American Mathematics Competitions