Proof: let's take it as granted log(n!) = O(n log n). It remains to show that log(n!) = Omega(n log n).

log(n!) = sum of log i for i from 1 to n

>= sum of log (n/2) for i from n/2 to n (dropping half the terms (all nonnegative) and under-approximating the rest; technically, we need to floor or ceil n/2, but it doesn't matter)

= (n/2) log (n/2)

= (n/2) (log n - log 2)

= (1/2) n log n - n log 2 / 2

Hopefully, I don't need to bother proving that c(n log n) + kn = Theta(n log n) for constants c and k. We've shown that log n! is Omega(n log n) and O(n log n), so we have log n! = Theta(n log n) by definition.

So, the root of this thread is wrong (and so is the parent). Furthermore, the fact that log(n!) is smaller than n log n by a constant is irrelevant when comparing log(n!) to Theta(n log n). What would not be meaningless is comparing formulae for the exact number of comparisons performed by different algorithms.