Problem: Which positive numbers x satisfy the equation (log3βx)(logxβ5)=log3β5?
Answer Choices:
A. 3 and 5 only
B. 3,5 and 15 only
C. only numbers of the form 5nβ
3m, where n and m are positive integers
D. all positive xξ =1
E. none of these
Solution:
For any fixed positive value of x distinct from one, let a=log3βx,b=logxβ5 and c=log3β5. Then x=3a,5=xb and 5=3c. These last equalities imply 3ab=3c or ab=c. Note that logxβ5 is not defined for x=1.