Statement
Every logarithm can be written as a division of two logarithms with a custom base:
$$ \log_a(b) = \frac{\log_c(b)}{\log_c(a)} $$
Proof
Let $ x $, $ y $ and $ z $ be logartihms and write them as exponentials.
$$ x = \log_a(b) \implies b = a^x $$
$$ y = \log_c(b) \implies b = c^y $$
$$ z = \log_c(a) \implies a = c^z $$
From $ b = a^x $ and $ b = c^y $ follows the following.
$$ a^x = c^y $$
Now substitute $ a = c^z $.
$$ (c^z)^x = c^y $$
Multiply the exponents and set them equal to eachother.
$$ c^{xz} = c^y $$
$$ xz = y $$
Divide both sides by $ z $ to isolate $ x $.
$$ x = \frac{y}{z} $$
Finally, re-substitue $ x $, $ y $ and $ z $.
$$ \log_a(b) = \frac{\log_c(b)}{\log_c(a)} $$