$\lim\limits_{n \to \infty}\frac {\log(n)}{\log(p_{n})}=1 $ where $p_{n}$ denotes the nth prime

Question

My professor today in class mentioned that the limit: $$\lim_{n \to \infty}\dfrac {\log(n)}{\log(p_{n})}=1$$I'm not sure if I agree since $p_{n}$ grows much faster than $n$. Was his statement correct? And if yes could someone help me understand the intuition behind this.

Answer 1

$$ \lim \limits_{n \to \infty} \frac{p_n}{n \ln n} =1 \implies \lim \limits_{n \to \infty} \frac{\ln n}{\ln p_n} =\lim \limits_{n \to \infty} \frac{\ln n}{\ln (n \ln n)} = \lim \limits_{n \to \infty} \frac{\ln n}{\ln n +\ln \ln n} = 1 $$