Problem:
Suppose x and y are positive real numbers such that
xy= 264 and (log2x)log2y= 27
What is the greatest possible value of log2y ?
Answer Choices:
A. 3
B. 4
C. 3+2
D. 4+3
E. 7
Solution:
Taking the base-2 logarithm of both equations gives
ylog2x=64 and (log2y)(log2log2x)=7
Now taking the base-2 logarithm of the first equation again yields
log2y+log2log2x=6
Let r=log2y and s=log2log2x. Then r+s=6 and rs=7. Solving this system yields
{log2y,log2log2x}∈{3−2,3+2}
Thus the greatest possible value of log2y is 3+2. This is achieved when x=223−2≈8.0095 and y=23+2≈21.3212.
The problems on this page are the property of the MAA's American Mathematics Competitions