A logarithm is the inverse operation to exponentiation. Given a base $b > 0$ and $b \not = 1$, the logarithm $\log_b(a)$ is the exponent $x$ you would raise $b$ to in order to get $a$.

Interchange of bases

Note this follows as if we let $\zeta = \log_c(b)$ then $b = c^\zeta$. So we have

$$b^y = (c^\zeta)^y = c^{y\zeta}$$

Lets equate this all to $x$ and use the definition above, on the left hand side we have

$$x = b^y \iff \log_b(x) = y$$

and on the right hand side

$$x = c^{\zeta y} \iff \zeta y = \log_c(x)$$

which combining these and substituting for $y$ gives

$$\log_c(b) \ast \log_b(x) = \zeta \log_b(x) = \zeta y = \log_c(x).$$

Natural logarithms

A lot of people will only talk about logarithms using the base $e$ instead of using the notation $\log_e(x)$ they may use the notation $\ln(x)$. This is sort of implied when people just write $\log(x)$. The reason for this is $\log_a(x)$ and $\log_b(x)$ only differ by a constant $\log_c(b)$, as above showed us - in some fields people don’t care about things up to scalar multiple.

Logs turn multiplication into addition

Logs are useful as they bring the rules of exponential powers to regular terms.

Which is proven by the fact that

$$e^Ae^B = e^{A+B}.$$

This by extension gives

$$\log(A^n) = n\log(A).$$

The inverse also holds

$$ \log(A) - \log(B) = \log(\frac{A}{B}) \ \mbox{ which is similar to } \ \frac{e^A}{e^B} = e^{A-B}.$$

Taylor expansion