Problem:
Which of the following is the value of log26+log36 ?
Answer Choices:
A. 1
B. log56
C. 2
D. log23+log32
E. log26+log36
Solution:
Recall that:
logb(uv)=logbu+logbv⋅logbu⋅logub=1. We use these properties of logarithms to rewrite the original expression:
log26+log36=(log22+log23)+(log32+log33)
=2+log23+log32
=(log23+log32)2
=(D)log23+log32
OR
First,
log26+log36=log2log6+log3log6=log3⋅log2log6⋅log3+log6⋅log2=log2⋅log3log6(log2+log3)
From here,
log2⋅log3log6(log2+log3)=log2⋅log3(log2+log3)(log2+log3)=log2⋅log3(log2)2+2⋅log2⋅log3+(log3)2
Finally,
log2⋅log3(log2)2+2⋅log2⋅log3+(log3)2=log2⋅log3(log2+log3)2
=log2⋅log3log2+log2⋅log3log3
=log3log2+log2log3
=log32+log23
Answer:
(D)log23+log32
The problems on this page are the property of the MAA's American Mathematics Competitions