1959 AMC 12 Problem 45

Attempt Problem 45 of the 1959 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1959 AMC 12 solutions, or check the answer key.

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45.

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.

Problem 45 in other years