Problem:
The solution to the equation log3x4=log2x8, where x is a positive real number other than 31 or 21, can be written as qp, where p and q are relatively prime positive integers. What is p+q?
Answer Choices:
A. 5
B. 13
C. 17
D. 31
E. 35
Solution:
By the change-of-base formula, the given equation is equivalent to
log3xlog4=log2xlog8log3+logx2log2=log2+logx3log22log2+2logx=3log3+3logxlogx=2log2−3log3logx=log274.
Therefore x=274, and the requested sum is 4+27=31.
OR
Changing to base-2 logarithms transforms the given equation into
log23x2=log22x32log22x=3log23xlog2(2x)2=log2(3x)3(2x)2=(3x)3,
so x=274, and the requested sum is 4+27=31.
The problems on this page are the property of the MAA's American Mathematics Competitions