How can I calculate limit of division of two logarithms

Question

How can I calculate $\lim\limits_{n \to \infty} \frac{\log_{a} n}{\log_{b} n}$. Where $a$ and $b$ are two integers.

Answer 1

Note that $$\frac{\log_a(n)}{\log_b(n)}=\log_a(b)$$ (see here).

Answer 2

$\lim\limits_{n \to \infty} \frac{\log_{a} n}{\log_{b} n}$, Now,if we change bases to e we get following expressions:

$\lim\limits_{n \to \infty}\frac{\ln n/\ln a}{\ln n/ \ln b} =\lim\limits_{n \to \infty} \frac{\ln b}{\ln a}=\frac{\ln b}{\ln a} $