Problem: Which positive numbers x satisfy the equation (log3​x)(logx​5)=log3​5?
Answer Choices:
A. 3 and 5 only
B. 3,5 and 15 only
C. only numbers of the form 5nâ‹…3m, where n and m are positive integers
D. all positive xî€ =1
E. none of these
Solution:
For any fixed positive value of x distinct from one, let a=log3​x,b=logx​5 and c=log3​5. Then x=3a,5=xb and 5=3c. These last equalities imply 3ab=3c or ab=c. Note that logx​5 is not defined for x=1.