Why is O(log(n)) coming equal to O(log(n!))?

Question

While solving the complexity of a code, I found it as O(log(n!)). I know that this can be proven equal to O(n*log(n)). However, can someone tell where this proof is going wrong?

Theorems used:

• log(ab) = log(a) + log(b)
• O(a+b) = O(max(a,b))

Proof

O(log(n!)) = O(log(n*(n-1)*(n-2)*...*2*1))

           = O(log(n) + log(n-1) + ... )

           = O(max(logn, log(n-1), ...))

           = O(log(n))

Can someone tell where I'm going wrong?

Answer 1

You cannot say O(log(n) + log(n-1) + ... ) = O(max(logn, log(n-1), ...))

This is only true for a constant number of summands. In your case the number depends on n.

Otherwise you could also proof

O(n)=O(1+1+1+1+1+...1) = O(max(1,1,1,...))= O(1)