# A calculus of the absurd

#### 9.2 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 $$p$$.

\begin{equation*} \log (wq) = \log (b^z b^p) \end{equation*}

We can then use one of the law of powers, that $$b^{x}b^{y} = b^{x + y}$$ 5151 This is explored above. to write that

\begin{equation*} \log (b^z b^p) = \log (b^{z+p}) \end{equation*}

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

\begin{equation*} \log (b^{z+p}) = z + p \end{equation*}

And from our earlier definitions of $$z=\log _b(w)$$ and $$p=\log _b(q)$$ we can say that 5252 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).

\begin{align} \label {product to sum log law} \log (wq) &= z + p \notag \\ &= \log _b(w) + \log _b(q) \end{align}