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.

A small $caveat$: "someone" is wrong: try, for example, with $N=2$; then
$$2\log 2=\log 2^2,$$
while
$$\log1+\log 2=\log 2!=\log 2.$$
For more details on the relationship between the 2 logarithms, I refer to the comments under the OP.
