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 $x > 2001^{2002}\ \mathrm{(B)}$ .

2004 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 15	Followed by Problem 17
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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2004 AMC 12A Problems/Problem 16

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