Asymptotic behavior of $n\sqrt n + n \log n$ & $\log_{100} n$
$\log_{100}n \in \mathcal{O}(log(n))$ since base change is a constant.


$\log n \in \mathcal{O}(n) $
$\log n \in \mathcal{O}(n^2) $
$\log n \in \mathcal{O}(n^3) $
$\log n \in \mathcal{O}(n^{3/2}) $
$\log n \not\in \mathcal{\omega}(n) $
$\log n \in \mathcal{o}(n \log n) $



This table gives the order;

First of all;

$$\lim_{n \rightarrow \infty} \frac{n \sqrt n}{n \log n} = \infty$$ by using this;


$f(n)= n\sqrt n + n\log n \in \mathcal{O}(n\sqrt n )$.


$ n\sqrt n \not\in \mathcal{O}(n)$
$n\sqrt n \in \mathcal{O}(n^2) $
$n\sqrt n \in \mathcal{O}(n^3) $
$n\sqrt n \in \mathcal{O}(n^{3/2}) $
$n\sqrt n \in \mathcal{\omega}(n) $
$n\sqrt n \not\in \mathcal{o}(n \log n) $,