How to calc $\log1+\log2+\log3+\log4+...+\log N$? How to calculate $\log1+\log2+\log3+\log4+...+\log N= log(N!)$? Someone told me that it's equal to $N\log N,$ but I have no idea why.

The integral 
$$\int_1^{N}\log(x) \; dx= N\log N - N +1$$
is a good approximation.
For $N=100$, the sum equals $361.3197940$ while the above expression gives $361.5170185.$
