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.