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2004 AMC 12A Problem 16

Problem 16 of 25IntermediateAlgebra

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

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Solution

The expression is defined if and only if log⁡2003(log⁡2002(log⁡2001x))>0,\log_{2003}(\log_{2002}(\log_{2001} x)) \gt 0, that is, log⁡2002(log⁡2001x)>1.\log_{2002}(\log_{2001} x) \gt 1. This holds if and only if log⁡2001x>2002,\log_{2001} x \gt 2002, which is equivalent to x>20012002.x \gt 2001^{2002}. Therefore c=20012002.c = 2001^{2002}. Thus, the correct answer is B.

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Concepts: logarithm · inequality

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.