Logistic function

Definition

The logistic function is a function with domain $\mathbb {R}$ and range the open interval $(0,1)$, defined as:

$x\mapsto {\frac {1}{1+e^{-x}}}$

Equivalently, it can be written as:

$x\mapsto {\frac {e^{x}}{e^{x}+1}}$

For this page, we will denote the function by the letter $g$.

We may extend the logistic function to a function $[-\infty ,\infty ]\to [0,1]$, where $g(-\infty )=0$ and $g(\infty )=1$.

Probabilistic interpretation

The logistic function transforms the logarithm of the odds to the actual probability. Explicitly, given a probability $p$ (strictly between 0 and 1)of an event occurring, the odds in favor of $p$ are given as:

${\frac {p}{1-p}}$

This could take any value in $(0,\infty )$

The logarithm of odds is the expression:

$\ln \left({\frac {p}{1-p}}\right)$

If $x$ equals the above expression, then the function describing $p$ in terms of $x$ is the logistic function.

Functional equations

Symmetry equation

The logistic function $g$ has the property that its graph $y=g(x)$ has symmetry about the point $(0,1/2)$. Explicitly, it satisfies the functional equation:

$g(x)+g(-x)=1$

Differential equation

The logistic function satisfies the condition:

$g'(x)=g(x)(1-g(x))$

Therefore, $y=g(x)$ is a solution to the autonomous differential equation:

${\frac {dy}{dx}}=y(1-y)$

The general solution to that equation is the function $y=g(x+C)$ where $C\in \mathbb {R}$. The initial condition $y=1/2$ at $x=0$ pinpoints the logistic function uniquely.

Key data

Item	Value
default domain	all of $\mathbb {R}$, i.e., all reals
range	the open interval $(0,1)$, i.e., the set $\{x\mid 0\leq x\leq 1\}$
derivative	the derivative is ${\frac {-e^{-x}}{(1+e^{-x})^{2}}}$. If we denote the logistic function by the letter $g$, then we can also write the derivative as $g'(x)=g(x)g(-x)=g(x)(1-g(x))$
second derivative	If we denote the logistic function by the letter $g$, then we can also write the derivative as $g'(x)=g(x)g(-x)=g(x)(1-g(x))(1-2g(x))$
logarithmic derivative	the logarithmic derivative is ${\frac {e^{-x}}{1+e^{-x}}}$ If we denote the logistic function by $g$, the logarithmic derivative is $g(-x)$
critical points	none
critical points for the derivative (correspond to points of inflection for the function)	$x=0$; the corresponding point on the graph of the function is $(0,1/2)$.
local maximal values and points of attainment	none
local minimum values and points of attainment	none
intervals of interest	increasing and concave up on $(-\infty ,0)$ increasing and concave down on $(0,\infty )$
horizontal asymptotes	asymptote at $y=0$ corresponding to the limit for $x\to -\infty$ asymptote at $y=1$ corresponding to the limit for $x\to \infty$
inverse function	inverse logistic function or log-odds function given by $x\mapsto \ln \left({\frac {x}{1-x}}\right)$