How do we prove $p_n\sim n\log(n\log(n))$ from the Prime Number Theorem?
Here is a sketchy derivation.
Note that
$$ n \log(n \log n) = n \log n + n \log \log n \sim n \log n $$
so we just need to prove the simpler statement that $p_n \sim n \log n$.
The formulas
$$ \pi(n) \sim \frac{n}{\log n} $$ $$ \pi(p_n) \sim n $$
imply
$$ \begin{align} n &\sim \frac{p_n}{\log p_n} \\ p_n &\sim n \log p_n \\ &\sim n \log(n \log p_n) \\ &\sim n (\log n + \log \log p_n) \\ &\sim n \log n \end{align}$$