Problem:
What value of x satisfies
log2x+log3xlog2x⋅log3x=2?
Answer Choices:
A. 25
B. 32
C. 36
D. 42
E. 48
Solution:
Observe that}
2=log2x+log3xlog2x⋅log3x=log3x1+log2x11=logx3+logx21=logx61=log6x.
It follows that x=62=(C)36.
OR
The given equation is equivalent to
log2x⋅log3x=2log2x+2log3x
Note that log3x=log23log2x, so
log2x⋅log23log2x=2log2x+2log23log2x
Multiplying both sides by log2xlog23 gives
log2x=2log23+2=log29+2
Then
x=2log29+2=2log29⋅22=9⋅4=(C)36.
The problems on this page are the property of the MAA's American Mathematics Competitions