# Difference between revisions of "2015 AMC 12A Problems/Problem 14"

## Problem

What is the value of $a$ for which $\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1$?

$\textbf{(A)}\ 9\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 36$

## Solution

We use the change of base formula to show that $$\log_a b = \dfrac{\log_b b}{\log_b a} = \dfrac{1}{\log_b a}.$$ Thus, our equation becomes $$\log_a 2 + \log_a 3 + \log_a 4 = 1,$$ which becomes after combining: $$\log_a 24 = 1.$$ Hence $a = 24$, and the answer is $\textbf{(D)}.$

~ pi_is_3.14

## Video Solution

Video starts at 1:04, uses the above solution: https://www.youtube.com/watch?v=OJyWHzjiu2A&t=44s

 2015 AMC 12A (Problems • Answer Key • Resources) Preceded byProblem 13 Followed byProblem 15 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 12 Problems and Solutions