2015 AMC 12A Problem 14 — Solution | LIVE by Po-Shen Loh

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2015 AMC 12A Problem 14

Below is the professionally curated solution for Problem 14 of the 2015 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2015 AMC 12A solutions, or check the answer key.

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Concepts:logarithm

Difficulty rating: 1730

14.

What is the value of $a$ for which $\dfrac{1}{\log_2 a} + \dfrac{1}{\log_3 a} + \dfrac{1}{\log_4 a} = 1?$

$9$

$12$

$18$

$24$

$36$

Solution:

By the change-of-base formula, $\dfrac{1}{\log_b a} = \log_a b.$ Therefore $1 = \log_a 2 + \log_a 3 + \log_a 4 = \log_a 24.$

It follows that $a = 24.$

Thus, the correct answer is D.

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