Log-odds (Logit)

Odds turn a probability into a ratio — a 75% chance of rain is odds of 3-to-1 (three times as likely to rain as not). Log-odds is simply the natural logarithm of those odds, and it is the scale on which logistic regression actually does its straight-line arithmetic.

The point of the log is to stretch a probability, which is trapped between 0 and 1, into a number that can run freely from minus infinity to plus infinity, so a linear equation behaves nicely. That is why a logistic-regression coefficient is a change in log-odds per unit of input — clean math, but not intuitive, which is exactly why people exponentiate it back into an odds ratio to talk about results.