2002 AMC 12A Problem 14

Attempt Problem 14 of the 2002 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 2002 AMC 12A solutions, or check the answer key.

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14.

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?

N \gt 1

N = 1

1 \lt N \lt 2

N = 2

N \gt 2

f(n) = \log_{2002} n^2 = 2\log_{2002} n

Combine into

N = \log_{2002}(11\cdot 13\cdot 14)^2

and evaluate the product inside

Using

\log a^2 = 2\log a

and adding logs,

\begin{aligned} N &= \log_{2002} 11^2 + \log_{2002} 13^2 \\ &\quad {}+ \log_{2002} 14^2 \\ &= \log_{2002}(11\cdot 13\cdot 14)^2. \end{aligned}

Since

11\cdot 13\cdot 14 = 2002,

this is

\log_{2002} 2002^2 = 2.

Thus, the correct answer is D.

Problem 14 in other years