Help with proving an equation factorial-time complexity

Question

I've been recently asked by one of my friends to prove an equation but still, I'm confused how to get it started tho.

log(n!)= θ(nlog(n))

Does anyone know how to help? I'll be very grateful if someone comes to reply to my issue.

Thanks in advance.

Answer 1

$$O(\log{(n!)})$$$$=O(\log{(n(n-1)(n-2)...(2)(1))})$$$$=O(\log{(n)}+\log{(n-1)}+\log{(n-2)}+...+\log{(2)}+\log{(1)})$$$$=O(n\log{(n))}$$ As $n$ logarithms are added, we have a worst case time complexity of $O(n\log{(n))}$.