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2002 AMC 12A Problems/Problem 14

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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{2}.

See Also

2002 AMC 12A (Problems • Resources)
Preceded by
Problem 13 		Followed by
Problem 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
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