6.57 If $\left(\log _{3} x\right)\left(\log _{x} 2 x\right)\left(\log _{2 x} y\right)=\log _{x} x^{2}$, then $y$ equals
(A) $\frac{9}{2}$.
(B) 9 .
(C) 18 .
(D) 27 .
(E) 81 .
(10th American High School Mathematics Examination, 1959)
[Solution] By the properties of logarithms, the given equation can be transformed as
$$
\frac{\lg x}{\lg 3} \cdot \frac{\lg 2 x}{\lg x} \cdot \frac{\lg y}{\lg 2 x}=\frac{\lg x^{2}}{\lg x},
$$

which simplifies to $\frac{\lg y}{\lg 3}=2$ or $\log _{3} y=2$, yielding $y=3^{2}=9$. Therefore, the correct choice is (B).