What is the value of a for which (1)/(\log_2 a) + (1)/(\log_3 a) + (1)/(\log_4 a) = 1?

2015 AMC 12A Problem 14 — Solution

2015 AMC 12A Problem 14

Attempt Problem 14 of the 2015 AMC 12A 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 2015 AMC 12A solutions, or check the answer key.

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14.

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

$9$

$12$

$18$

$24$

$36$

Answer: D

Concepts:logarithm

Difficulty rating: 1730

Solution:

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

It follows that $a = 24.$

Thus, the correct answer is D.

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