Sum of logs in the form of $1\log1 +2\log2 +\ldots +n\log n$

The sum of this series is an approximation of (and is approximated by) the integral $$ \int_1^{n+1} x\log(x)\,dx = \Theta(n^2\log(n)) $$ So, your sum of logs will be $\Theta(n^2\log(n))$.