number theory - Estimate how much large is $lfloor log{n!rfloor}$

Is there a simple method to find or estimate how large $$\lfloor \lg{n!\rfloor}$$ is ? I'd like to find (or estimate) how much digits $2017!^{2017}$ has, or how much is big that number .

I tried for some little number and general form of $$n!^{n}=\overline{a_k...a_4a_3a_2a_1} \\ k=?$$ so
I took logarithm of $n!^{n} \mapsto n\log(n!) $.

It is easy when $n$ is little number , but when go for a large number ...what we can do ?
My question can be translate as $$n\sum_{i=1}^{n}\log(i)= ?$$

Answer

Stirling's approximation says that
$$
n!\approx \sqrt{2\pi n}\left(\frac ne\right)^n
$$
Taking logarithms on both sides should be easy enough.