Log(s) = logarithms

This is a funny one because I suspect that people who liked school maths (or more than school maths) know this and I fear that those who didn’t will find it a bit allergenic. However, I have just added odds or odds ratios to this glossary and “log odds” is a term you may encounter in research papers so here we go.

Details #

The logarithm of a number is (usually) the power you have to raise 10 to in order to get to the number. So if you need the log (logarithm) of 10, it’s 1, the log of 100 is 2 and if you need the log of 1,000 it’s 3 because:

$$10^{1} = 10$$

$$10^{2} = 100$$

and

$$10^{3} = 1000$$

Equally the log of 0.1 is -1 because:

$$10^{-1} = 0.1$$

So what? Well …
- One useful effect is that “taking log(arithms)” can reduce huge and tiny numbers to less a more compact form: $log(1,000,000) = 6$ and $log(0.000001) = -6$.
- That gives us “scientific notation” so if you see
  “p = 1.73^-6” or “p = 1.73e-6”
  those are shorthand for
  “p = 0.00000173”,
  because $1.73 * 10^{-6}=0.00000173$. That’s a not uncommon way of expressing small p values in our field though I think it only fairly rarely makes it into papers.
- It reflects the way that human perceptual systems reflect some things so we experience length in linear fashion: a 2m gap between chairs looks about twice a 1m gap but we experience loudness in (roughly) logarithmic scale: a noise that sounds about twice as loud as another may involve about 10x as much energy in its production and one that sounds 4x the loudness of the first may involve about 100x as much energy as the first. (I oversimplify but this is partly why huge speaker stacks to blast open air rock performances need a lot of power to drive them!)
- Logarithms convert multiplicative relationships to ones of addition so as:
  1,000*100 = 100,000
  we have
  log(1,000) + log(100) = log(100,000) = 3 + 2 = 5 and log(100,000) is 5. This can be very useful when dealing with probabilities and odds as they generally have multiplicative relationships so the probability of getting two heads in a row tossing a fair coin is .5*.5 = .25. It’s also useful for analysing “interactions”: situationas in which the joint effect of two variables on a third is not just additive.
- And that last attribute is why you will often see odds ratios expressed as “log odds”. See the entry about odds ratios to see a plot using log odds.

One caveat: all the way through this I have been talking of “logarithms to the base 10” or $\text{log}_{10}$ to give them their full name and notation. It is just possible that you will see “natural log(arithms)” mentioned. These are $\text{log}_{e}$ where $e\cong 2.718282$. The “log odds” you may see in some papers are these, not logarithms to base 10 but all the principles are the same. If you find yourself there and want to learn more, find a statistician or mathematician!