This is pretty simple and I THINK I know the answer to the question, but I don't know how to prove it formally. Below follows the question.
Question. Compare the functions $f(n) = \frac{n^2}{\log(n)}$ and $g(n) = 3^{\log(n)}$ in terms of rate growth.
My attempt. I have tried to compute the limit $\lim_{n \rightarrow +\infty} \frac{f(n)}{g(n)}$ but this doesn't seem easy by hand. Using symbolab, it says it can't compute the limit. Obviously my intuition points towards $f(n) << g(n)$ just because $g(n)$ is an exponential function, and normally it grows faster than the polynomial ones, but I don't know how to prove this for the functions above.
Besides what I've shown above, I also tried taking $\log$ in both the functions, which would origin $2\log(n) - \log(\log(n))$ for $f(n)$ and $\log(n)\log(3)$ for $g(n)$ and the limit between these two is a constant which leads me to believe I cant really use this to comparate both the functions (Since my intuition tells me $f(n) << g(n)$, the limit I stated would be a contradiction).
So here come my two questions:
Can I use the $\log$ property above to compare two functions? I.e., does $f(n) << g(n) \Leftrightarrow \log(f(n)) << \log(g(n))$ ? More generally, is the comparation (of rates of growth) between $f(n)$ and $g(n)$ equivallent to comparing the rates of growth of $\log(f(n))$ and $\log(g(n))$ ?
How would one compare the functions from the question formally?
Thanks for any help in advance.