[summary: The logistic function is a [-sigmoid] function that maps the real numbers to the unit interval $~$(0, 1)$~$ using the formula $~$\displaystyle f(x) = \frac{1}{1 + e^{-x}}$~$ or variants of this formula.]

The logistic function is a [-sigmoid] function that maps the real numbers to the unit interval $~$(0, 1)$~$ using the formula $~$\displaystyle f(x) = \frac{1}{1 + e^{-x}}$~$.

More generally, there exists a [family_of_functions family] of logistic functions that can be written as $~$\displaystyle f(x) = \frac{M}{1 + \alpha^{c(x_0 - x)}}$~$, where:

$~$M$~$ is the upper bound of the function (in which case the function maps to the interval $~$(0, M)$~$ instead). When $~$M = 1$~$, the logistic function is usually being used to calculate some Probability or Proportion of a total.

$~$x_0$~$ is the inflection point of the curve, or the value of $~$x$~$ where the function's growth stops speeding up and starts slowing down.

$~$\alpha$~$ is a variable controlling the steepness of the curve.

$~$c$~$ is a scaling factor for the distance.

## Applications

The logistic function is used to model growth rates of populations in an ecosystem with a limited carrying capacity.

The inverse logistic function (with $~$\alpha = 2$~$) is used to convert a probability to log-odds form for use in Bayes' rule.

The logistic function (with $~$\alpha = 10$~$ and $~$c = 1/400$~$) is used to calculate the expected probability of a player winning given a specific difference in rating in the Elo rating system.