TITLE: $\prod _{k=2}^{n} {\log k}$ is big-$O$ of what?
QUESTION [5 upvotes]: $$\prod _{k=2}^{n} {\log k}$$
is a big-$O$ of what? 
I can see it $O(n!)$ but is there a tighter solution?

REPLY [7 votes]: If your product is $P(n)$, then $\log P(n) = \sum_{k=2}^n \log \log k
\le n \log \log n$, so $P(n) \le \exp((n-1) \log \log n) = (\log n)^{n-1}$.