This seems a bit weird. I thought that $\frac{n^3}{\log(n)}$ is tight bounded by $n^3$ since its upper bound should also be $n^3$ and it's lower bound can be for example $\frac{1}{2}$
However, I realized that as $n$ grows larger, $\frac{n^3}{2}$ would actually surpass $\frac{n^3}{\log(n)}$. So what I want to know is if there exists a tight-bound for $\frac{n^3}{\log(n)}$, or for that case for any rational expression and the behaviours for bounds on rational expressions. (since it seems like any constant will eventually be so little compared to a moving variable).
Furthermore, does this mean that the lower bound for $\frac{n^3}{\log(n)}$ is $n^2$?
$\log{n}$ grows more slowly than any positive power of $n$ (you can check this by differentiating, for example). Therefore for every $\epsilon > 0$, then for sufficiently large $n$,
$$ \frac{n^3}{\log{n}} > n^{3 - \epsilon}$$
This is why when people look at computational complexity, logarithmic terms are sometimes not considered bad.