1953 AMC 12 Problem 39

Attempt Problem 39 of the 1953 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 1953 AMC 12 solutions, or check the answer key.

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39.

The product,

\log_a b\cdot\log_b a

is equal to:

1

a

b

ab

none of these

Answer: A

Concepts:logarithms change of base

Difficulty rating: 1180

Small hint:

Use the change-of-base formula on both logarithms

Big hint:

The two resulting fractions are reciprocals

Solution:

For permissible bases and arguments,

\begin{aligned} \log_a b\cdot\log_b a &=\frac{\log b}{\log a}\\ &\quad{}\cdot\frac{\log a}{\log b}\\ &=1. \end{aligned}

Thus, the correct answer is A.

Problem 39 in other years