Problem:
What value of x satisfies
log2​x+log3​xlog2​x⋅log3​x​=2?
Answer Choices:
A. 25
B. 32
C. 36
D. 42
E. 48
Solution:
Observe that}
2​=log2​x+log3​xlog2​x⋅log3​x​=log3​x1​+log2​x1​1​=logx​3+logx​21​=logx​61​=log6​x.​
It follows that x=62=(C)36​.
OR
The given equation is equivalent to
log2​x⋅log3​x=2log2​x+2log3​x
Note that log3​x=log2​3log2​x​, so
log2​x⋅log2​3log2​x​=2log2​x+2log2​3log2​x​
Multiplying both sides by log2​xlog2​3​ gives
log2​x=2log2​3+2=log2​9+2
Then
x=2log2​9+2=2log2​9⋅22=9⋅4=(C)36​.
The problems on this page are the property of the MAA's American Mathematics Competitions