What is the value of the [expression](https://artofproblemsolving.com/wiki/index.php/Expression)
\[\frac{1}{\log_2 100!} + \frac{1}{\log_3 100!} + \frac{1}{\log_4 100!} + \cdots + \frac{1}{\log_{100} 100!}?\]
$\mathrm{(A)}\ 0.01 \qquad\mathrm{(B)}\ 0.1  \qquad\mathrm{(C)}\ 1 \qquad\mathrm{(D)}\ 2 \qquad\mathrm{(E)}\ 10$

Solution 1
By the change-of-base formula, 
\[\log_{k} 100! = \frac{\log 100!}{\log k}\]  
Thus (because if $x=y$, then $\frac{1}{x}=\frac{1}{y}$)
\[\frac{1}{\log_k 100!} = \frac{\log k}{\log 100!}\]
Thus the sum is 
\[\left(\frac{1}{\log 100!}\right)(\log 2 + \log 3 + \cdots + \log 100) = \frac{1}{\log 100!} \cdot \log 100! = 1 \Rightarrow \mathrm{(C)}\]

Solution 2
Since $1=\log_{k} k$,
\[\frac{1}{\log_{k}100!}=\frac{\log_{k}k}{\log_{k}100!}=\log_{100!} k\]
We add:
\[\log_{100!} 1 +\log_{100!} 2 +\log_{100!} 3 +\cdots + \log_{100!} 100=\log_{100!}100!=1 \Rightarrow \mathrm{(C)}\]
empty