2002 AMC 12A Problems/Problem 14

Problem

For all positive integers $n$ , let $f(n)=\log_{2002} n^2$ . Let $N=f(11)+f(13)+f(14)$ . Which of the following relations is true?

$\text{(A) }N<1 \qquad \text{(B) }N=1 \qquad \text{(C) }1<N<2 \qquad \text{(D) }N=2 \qquad \text{(E) }N>2$

Solution

First, note that $2002 = 11 \cdot 13 \cdot 14$ .

Using the fact that for any base we have $\log a + \log b = \log ab$ , we get that $N = \log_{2002} (11^2 \cdot 13^2 \cdot 14^2) = \log_{2002} 2002^2 = \boxed{(D) N=2}$ .

Video Solution

https://youtu.be/YUdemyBf2yI

2002 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
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All AMC 12 Problems and Solutions

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2002 AMC 12A Problems/Problem 14

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