The set of all real numbers x for which log_2004(log_2003(log_2002(log_2001 x))) is defined is {x x > c}. What is the value of c?

2004 AMC 12A Problem 16

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16.

The set of all real numbers

x

for which

\small \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}}

Work from the outside in: each logarithm requires its argument to be positive

\log_{2004}(\cdots)

is defined only when

\log_{2003}(\cdots) \gt 0,

i.e. the next argument exceeds

1

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.