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

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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

\dfrac{1}{\log_b a} = \log_a b

The sum becomes

\log_a(2\cdot 3\cdot 4)

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.