Problem:
Let x,y and z all exceed 1 and let w be a positive number such that
logxw=24,logyw=40 and logxyzw=12
Find logz⋅w.
Solution:
Converting each of the given logarithms into exponential form gives x24=w,y40=w,(xyz)12=w. It follows that
z12=x12y12w=w1/2w3/10w=w1/5
Thus w=z60 and logzw=60.
The problems on this page are the property of the MAA's American Mathematics Competitions