1965 AHSME Problems/Problem 31

Problem

The number of real values of $x$ satisfying the equality $(\log_ax)(\log_bx) = \log_ab$ , where $a > 0, b > 0, a \neq 1, b \neq 1$ , is:

$\textbf{(A)}\ 0 \qquad \textbf{(B) }\ 1 \qquad \textbf{(C) }\ 2 \qquad \textbf{(D) }\ \text{a finite integer greater than 2}\qquad \textbf{(E) }\ \text{not finite}$

Solution

From the properties of logarithms, we know that $(\log_ax)(\log_xb)=\log_ab$ , and $\log_xb=\frac{1}{\log_bx}$ . Thus, to satisfy the given equation, we are looking for values of $x$ when $\log_bx=\frac{1}{\log_bx}$ , or when $\log_bx=\pm 1$ . This happens either when $x=b$ or $x=\frac{1}{b}$ . Thus we have $2$ solutions, consistent with answer choice $\fbox{\textbf{(C)}}$ .

1965 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 30	Followed by Problem 32
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1965 AHSME Problems/Problem 31

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