Problem:
If log(xy3)=1 and log(x2y)=1, what is log(xy) ?
Answer Choices:
A. −21
B. 0
C. 21
D. 53
E. 1
Solution:
We have
1=log(xy3)=logx+3logy and 1=log(x2y)=2logx+logy
Solving yields logx=52 and logy=51. Thus
log(xy)=logx+logy=53
The given equations imply that xy3=10=x2y. Thus
y=x210 and x(x210)3=10
It follows that x=102/5 and y=101/5, so log(xy)=log(103/5)=3/5.
OR
Since log(xy3)=log(x2y), we have xy3=x2y, so x=y2. Hence
1=log(xy3)=log(y5)=5logy, and logy=51
So log(xy)=log(y3)=3/5.
The problems on this page are the property of the MAA's American Mathematics Competitions