Problem:
Let x,y and z all exceed 1 and let w be a positive number such that
logx​w=24,logy​w=40 and logxyz​w=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 logz​w=60​.
The problems on this page are the property of the MAA's American Mathematics Competitions