# Difference between revisions of "1975 AHSME Problems/Problem 19"

## Problem 19

Which positive numbers $x$ satisfy the equation $(\log_3x)(\log_x5)=\log_35$?

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

## Solution

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

## See Also

 1975 AHSME (Problems • Answer Key • Resources) Preceded byProblem 18 Followed byProblem 20 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 All AHSME Problems and Solutions

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