8.2 Logarithms

These seem scary at first, but they’re not actually too bad.

A logarithm has a "base", and a "power". When $\log_{a}(b)$ is written, it means "what needs to be raised to the power of $a$ to get $b$?" For example, $log_{2}(8)=3$, as $2^{3}=8$.

The definition of a logarithm is that $z=\log_{b}(w)$ if and only if $w=b^{z}$. From here, we can prove a bunch of facts about the logarithm function.

For example, if we let $z=\log_{b}(w)$ and $p=\log_{b}(q)$ then we can then express $\log(wq)$ in terms of $z$ and $p$.

We can then use one of the law of powers, that $b^{x}b^{y}=b^{x+y}$ ²² 2 This is explored above. to write that

After this, we can use the definition of the $\log$ function to simplify the right-hand side of the previous equation.

And from our earlier definitions of $z=\log_{b}(w)$ and $p=\log_{b}(q)$ we can say that ³³ 3 This is particularly powerful because it means that we can write any multiplication as a sum (and there’s a lot more algebra that can be applied to sums than products).