smc

For all real numbers $a,b,$ and $c$ such that $b>1,$ note that: 1. ${\log }_{b}a$ is defined if and only if $a>0.$ 2. ${\log }_{b}a>c$ if and only if $a>b^c.$ Therefore, we have $$\begin{array}{rl}{\log }_{2004}({\log }_{2003}({\log }_{2002}({\log }_{2001}x)))\text{ is defined}& ⟹{\log }_{2003}({\log }_{2002}({\log }_{2001}x))>0\\ & ⟹{\log }_{2002}({\log }_{2001}x)>1\\ & ⟹{\log }_{2001}x>2002\\ & ⟹x>2001^{2002},\end{array}$$ from which $c=\boxed{2001^{2002}}.$