2004 AMC 12A Problem 16

Below is the professionally curated solution for Problem 16 of the 2004 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2004 AMC 12A solutions, or check the answer key.

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Concepts:logarithm inequality

Difficulty rating: 1660

16.

The set of all real numbers $x$ for which $\log_{2004}(\log_{2003}(\log_{2002}(\log_{2001} x)))$ is defined is $\{x \mid x \gt c\}.$ What is the value of $c?$

$0$

$2001^{2002}$

$2002^{2003}$

$2003^{2004}$

$2001^{2002^{2003}}$

Solution:

The expression is defined if and only if $\log_{2003}(\log_{2002}(\log_{2001} x)) \gt 0,$ that is, $\log_{2002}(\log_{2001} x) \gt 1.$

This holds if and only if $\log_{2001} x \gt 2002,$ which is equivalent to $x \gt 2001^{2002}.$

Therefore $c = 2001^{2002}.$

Thus, the correct answer is B.

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