8.2 Logarithms

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

A logarithm has a "base", and a "power". When loga(b)\log_{a}(b) is written, it means "what needs to be raised to the power of aa to get bb?" For example, log2(8)=3log_{2}(8)=3, as 23=82^{3}=8.

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

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


We can then use one of the law of powers, that bxby=bx+yb^{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\log function to simplify the right-hand side of the previous equation.


And from our earlier definitions of z=logb(w)z=\log_{b}(w) and p=logb(q)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).

log(wq)\displaystyle\log(wq) =z+p\displaystyle=z+p
=logb(w)+logb(q)\displaystyle=\log_{b}(w)+\log_{b}(q) (8.3)