AHSME 1975 Problem 19

Which positive numbers $x$ satisfy the equation $(\log_3x)(\log_x5)=\log_35$? $\textbf{(A)}\ 3 \text{ and } 5 \text{ only} \textbf{(B)}\ 3, 5, \text{ and } 15 \text{ only} \\ \textbf{(C)}\ \text{only numbers of the form } 5^n \cdot 3^m, \text{ where } n \text{ and } m \text{ are positive integers} \\ \textbf{(D)}\ \text{all positive } x \neq 1 \textbf{(E)}\ \text{none of these}$