I need to find tight bound for $\Theta\big(n\cdot \lg n\cdot\lg\frac{n}{\lg n}\big)$, ($\Theta$) bound specifically.
I thought doing a limit: $$ \lim_{n\to \infty} \frac{n}{\lg n}=\lim_{n\to \infty} \frac{1}{1\over n}=\infty\implies\lg n=o(n)\implies\\\Theta\bigg(n\cdot \lg n\cdot\lg\frac{n}{\lg n}\bigg)=O(n\lg^2 n) $$
So I can get only an upper bound. In addition I wonder if there's a simpler way to do this because of logarithmic identity: $$ \Theta\bigg(n\cdot \lg n\cdot\lg\frac{n}{\lg n}\bigg)=\Theta(n\cdot \lg n\cdot(\lg n-\lg(\lg n))) $$ but I don't know how to go about the minus when determining a bound.