Estimation of $x$ if $x! = N^{\log N}$

Question

If $x! = N^{\log N}\;,$ How can I estimate $x$ in terms of $N$?

Answer 1

If we had an approximation for the inverse gamma function $\Gamma^{-1}$, we could apply it to both sides to get $$x \approx \Gamma^{-1} \left( N^{\log{N}} \right)$$

An approximation of $\Gamma^{-1}$ can be found here.