Is this true: $(n \log n)^2 \in \Theta(\frac{n^3}{\log n})$

Question

I have to decide whether $(n \log n)^2 \in \Theta(\frac{n^3}{\log n})$ is true or false.

I can't really find a good approach, I'd assume it's wrong, but I can't prove it.

Answer 1

if $(n \log n)^2 \in \Theta(\frac{n^3}{\log n})$ there is a constant $C>0$ such that $$n^2\log^2n\ge \frac{Cn^3}{\log n}$$ for $n$ sufficiently large. This simplifies to $$\log^3n\geq Cn$$ which is false.