Track / Stage 4 / 142 of 340 #402 of 1964

Problem 402

AMC 12 late, AIME early

Algebra Difficulty 4.8 Multiple choice

$8 \cdot 24$ satisfies the equation $\left(\log _{3} x\right)\left(\log _{x} 5\right)=\log _{3} 5$ for the positive number $x$

Pick one

Official solution

[Solution] If $x>0$ and $x \neq 1$ , let $a=\log _{3} x, \quad b=\log _{x} 5$ and $c=\log _{3} 5$ , then $x=3^{a}, \quad 5=x^{b}$ and $5=3^{c}$ .

Thus $3^{a b}=3^{c}$ or $a b=c$ .

That is, when $x \neq 1$ , $a b=c$ .
Therefore, the answer is $(D)$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.