Which case of the Master theorem applies to the recurrence $T(n)= 100T(n/99)+\log(n!)$?

Question

How to use the Master theorem to solve $T(n)= 100T(n/99)+\log(n!)$?

I was given this question, and I can't figure out which case of the master theorem goes here. Thanks for your suggestions.

Answer 1

Recall that: $$ \log(n!) \leq \log(n^n) = n\log n = O(n^k) $$ for any $k > 1$. Thus, since $\log(n!) = O(n^{(\log_{99}100) - \epsilon})$ for any $\epsilon \in (0, (\log_{99} 100) - 1)$, it follows by Case $1$ of the Master Theorem that: $$ T(n) = \Theta(n^{\log_{99} 100}) $$