Why does $\ln x / \ln b = \log_b x$? 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.

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,
$$ \log_b x = \frac{\ln x}{\ln b}.$$
