I'm doing some Java code. As far as I can tell, Java only has functions that do natural log and base $10$ log. I have a requirement to specify the base. I've seen that doing $\ln x/ \ln b$ is the same as $\log_b x$.
I've done this and it seems to work. I'd like to better understand why though.
$\endgroup$ 11 Answer
$\begingroup$Let $y = \log_b x$. This is the same as $x = b^y$.
If you take the natural logarithm on both sides you get $$\ln x = \ln b^y = y \ln b$$ so that $y = \dfrac{\ln x}{\ln b}$. That is, $$ \log_b x = \frac{\ln x}{\ln b}.$$
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