2004 AMC 12A Problem 16

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.