2004 AMC 12A Problems/Problem 16

Problem

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 $c$?

$\text {(A)} 0\qquad \text {(B)}2001^{2002} \qquad \text {(C)}2002^{2003} \qquad \text {(D)}2003^{2004} \qquad \text {(E)}2001^{2002^{2003}}$

Solution

We know that the domain of $\log_k n$, where $k$ is a constant, is $n > 0$. So $\log_{2003}(\log_{2002}(\log_{2001}{x})) > 0$. By the definition of logarithms, we then have $\log_{2002}(\log_{2001}{x})) > 2003^0 = 1$. Then $\log_{2001}{x} > 2002^1 = 2002$ and $x > 2001^{2002}\ \mathrm{(B)}$.

See also

2004 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 15
Followed by
Problem 17
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All AMC 12 Problems and Solutions
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