8.25 Let $a, b$ be positive constants not equal to 1. Then the number of real numbers $x$ that satisfy the equation $\left(\log _{a} x\right)\left(\log _{b} x\right)$ $=\log _{a} b$ is
(A) 0.
(B) 1.
(C) 2.
(D) An integer greater than 2.
(E) Not a finite number.
(16th American High School Mathematics Examination, 1965)
[Solution 1] Let $\log _{b} x=y$, then $x=b^{y}$. From the given equation, we have $\left(\log _{a} b^{y}\right) y=\log _{a} b$, which is $y^{2} \log _{a} b=\log _{a} b$, $\therefore y^{2}=1$, yielding $y= \pm 1$. Thus, $x=b$ or $b^{-1}$.
Therefore, the answer is $(C)$.
[Solution 2] Let $\log _{a} x=y$, then $z=a^{y}$. From the given equation, we have $y \log _{b} a^{y}=\log _{a} b$, which is $y^{2}=\left(\log _{a} b\right)^{2}$,

yielding $y= \pm \log _{a} b$, thus $x=b$ or $b^{-1}$.
Therefore, the answer is $(C)$.