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

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Hint: the elements of the sequence do not actually depend on $n$... –  Henning Makholm Oct 7 '11 at 15:19
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2 Answers 2

up vote 3 down vote accepted

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

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$\lim\limits_{n \to \infty} \frac{\log_{a} n}{\log_{b} n}$, 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} $

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