The set of all real numbers $x$ for which
\[\log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x})))\]
is defined is $\{x\mid x > c\}$. What is the value of $c$?
$\textbf {(A) } 0\qquad \textbf {(B) }2001^{2002} \qquad \textbf {(C) }2002^{2003} \qquad \textbf {(D) }2003^{2004} \qquad \textbf {(E) }2001^{2002^{2003}}$
For all real numbers $a,b,$ and $c$ such that $b>1,$ note that:


$\log_b a$ is defined if and only if $a>0.$
$\log_b a>c$ if and only if $a>b^c.$

Therefore, we have
\begin{align*} \log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x}))) \text{ is defined} &\implies \log_{2003}(\log_{2002}(\log_{2001}{x}))>0 \\ &\implies \log_{2002}(\log_{2001}{x})>1 \\ &\implies \log_{2001}{x}>2002 \\ &\implies x>2001^{2002}, \end{align*}
from which $c=\boxed{\textbf {(B) }2001^{2002}}.$
~Azjps ~MRENTHUSIASM