I was wondering how people go about showing the proofs with orders of growth? Currently, I have the following functions and I know what order they go in, but I'm not sure how to prove them. I simply put them into a graphing application I saw online and the graph itself obviously showed the growth.
$\begin{align} & 2^{\sqrt{\log n}}\\ & 2^n\\ & n^{4/3}\\ & n \log^3 n\\ & n^{ \log n}\\ & 2^{2^n}\\ & 2^{n^2} \end{align}$
For example, I know that $2^{2^n}$ has the largest growth rate, with $2^{n^2}$ coming in second. I know this because if I plug the value of $5$ it turns into $2^{32}$ vs $2^{25}$, $6$ is $2^{64}$ vs $2^{36}$. But, is that the proof itself?
As far as the odd ones out, I have no idea how I could show a proof comparison of $n^{4/3}$ vs $n\log^3 $n vs $2^n$, etc.