Asymptotic behavior of $n\sqrt n + n \log n$ & $\log_{100} n$ I have the following two functions


$f(n) = n\sqrt n + n \log n$
$\log_{100} n$


And I need to classify them into the followings:


$O(n)$, and/or
$O(n^2)$, and/or
$O(n^3)$, and/or
$O(n^{1.5})$, and/or
$\omega(n)$, and/or
$o(n\log n)$ (small o)
$\theta(n^{1.5})$, 


Regarding the $\theta(n^{1.5})$, I am grasping it's idea newly, where do these functions set in. Also, does the function $f(n) =n^{0.9999}$ belong to $\theta(n^{1.5})$.

I would appreciate your feedback, as from your answers I am building up the knowledge for these.
$\log_{100}n \in \mathcal{O}(log(n))$ 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) $,