Question

If $\begin{aligned} &(\log_3x)(\log_x2x)(\log_{2x}y)\\ &\qquad=\log_xx^2, \end{aligned}$ then $y$ equals:

$\dfrac92$

$9$

$18$

$27$

$81$

Answer: B

Concepts:logarithm telescoping

Difficulty rating: 1280

Small Hint:

Use $(\log_a b)(\log_b c)=\log_a c$ twice

Big Hint:

Simplify the entire left side to $\log_3y$

Solution:

The logarithms telescope: $\begin{aligned} &(\log_3x)(\log_x2x)(\log_{2x}y)\\ &\qquad=\log_3y. \end{aligned}$ Also $\log_xx^2=2,$ so $\log_3y=2$ and $y=3^2=9.$

Therefore, the correct answer is B.

1959 AMC 12 Problem 45

Answer: B

Difficulty rating: 1280

Small Hint:

Big Hint:

Solution:

Problem 45 in Other Years