Which positive numbers $x$ satisfy the equation $(\log_3x)(\log_x5)=\log_35$? 
$\textbf{(A)}\ 3 \text{ and } 5 \text{ only} \qquad  \textbf{(B)}\ 3, 5, \text{ and } 15 \text{ only} \qquad \\ \textbf{(C)}\ \text{only numbers of the form } 5^n \cdot 3^m, \text{ where } n \text{ and } m \text{ are positive integers} \qquad \\ \textbf{(D)}\ \text{all positive } x \neq 1 \qquad  \textbf{(E)}\ \text{none of these}$
By the change-of-base formula, we can simplify the left side of the equation: $(\log_3x)(\log_x5) = (\frac{\log_x}{\log_3})(\frac{\log_5}{\log_x}) = \frac{\log_5}{\log_3}$.
We see that this in fact simplifies to $\log_35$, which will always equal the right side of the equation, since they are the same exact expressions.
But we have to be careful because $x \neq 1$. Plugging in $x=1$, the left side would equal $(\log_31)(\log_x5) = 0 \cdot \log_x5 = 0$, and $\log_35$ definitely does not equal $0$.
Besides $1$, $x$ can take on any positive value, and the equation would work. Therefore, the answer is $\boxed{\textbf{(D)}\ \text{all positive } x \neq 1}$ ~[jiang147369](https://artofproblemsolving.comhttps://artofproblemsolving.com/wiki/index.php/User:Jiang147369)