Odds, logs and log odds

5.3. Odds, logs and log odds#

  • What are ‘odds’?

  • If the probability of something is 50-50, what are the odds?

  • What’s a “log”?

  • Can you express \(10^3\) as a log?

  • Let’s try this one, \(\log_2{16} = x\). What is \(x\)?

On Canvas, you’ll find additional materials on logs, if you need them.

In our examples above, we have seen log base 2 and log base 10 (these are the easiest to work with for working out in your head). Instead of using 2 or 10, the logit function in logistic regression uses the “natural logarithm” or \(\log_e\) where the base is the constant \(e\) which is much used in maths and statistics. \(e\) has a value of 2.718 and is known as Euler’s number or Euler’s constant. You don’t need to worry too much about why this is the way it is done, or where Euler’s constant comes from, but this is what is going on in our logarithmic transformation. (For the curious, there is an easy video on Canvas about Euler’s constant). The natural logarithm is often written as ln.

  • Can you figure out what the answer to this one would be?

\[ \log_e{1} = x \]