# Prove $\frac{n}{3}\log\ n\ - \frac{n}{2}\log \sqrt{n}$ is $\Omega(n\log\ n)$

Prove: $$\frac{n}{3}\log n - \frac{n}{2}\log \sqrt{n}$$ is $$\Omega(n\ \log n)$$.

I'm trying to use the Big-Omega definition, what I reached so far is:

Let

$$f(n)=\frac{n}{3}\log n - \frac{n}{2}\log \sqrt{n}$$

$$g(n) = (n\log n)$$

Then, $$f(n)$$ is in $$\Omega(g(n))$$ if there is $$c>0$$ and $$n_0 > 0$$ such that $$f(n) \geq c \times g(n)$$ for some $$n\geq n_0$$

How do I continue from here?

$$\frac{n}{3}\log\ n\ - \frac{n}{2}\log \sqrt{n} = \frac{n}{3}\log\ n\ - \frac{n}{4}\log {n} = \frac1{12}\, \left(n\log\ n\right)$$