Logistic function
Contents
Definition
The logistic function is a function with domain and range the open interval
, defined as:
Equivalently, it can be written as:
Yet another form that is sometimes used, because it makes some aspects of the symmetry more evident, is:
For this page, we will denote the function by the letter .
We may extend the logistic function to a function , where
and
.
Probabilistic interpretation
The logistic function transforms the logarithm of the odds to the actual probability. Explicitly, given a probability (strictly between 0 and 1) of an event occurring, the odds in favor of
are given as:
This could take any value in
The logarithm of odds is the expression:
If equals the above expression, then the function describing
in terms of
is the logistic function.
Key data
Item | Value |
---|---|
default domain | all of ![]() |
range | the open interval ![]() ![]() |
derivative | the derivative is ![]() If we denote the logistic function by the letter ![]() ![]() |
second derivative | If we denote the logistic function by the letter ![]() ![]() |
logarithmic derivative | the logarithmic derivative is ![]() If we denote the logistic function by ![]() ![]() |
antiderivative | the function ![]() |
critical points | none |
critical points for the derivative (correspond to points of inflection for the function) | ![]() ![]() |
local maximal values and points of attainment | none |
local minimum values and points of attainment | none |
intervals of interest | increasing and concave up on ![]() increasing and concave down on ![]() |
horizontal asymptotes | asymptote at ![]() ![]() asymptote at ![]() ![]() |
inverse function | inverse logistic function or log-odds function given by ![]() |
Differentiation
First derivative
Consider the expression for :
We can differentiate this using the chain rule for differentiation (the inner function being and the outer function being the reciprocal function
. We get:
Simplifying, we get:
We can write this in an alternate way that is sometimes more useful. We split the expression as a product:
The first factor on the right is , and the second factor is
, so this simplifies to:
Second derivative
Using the expression
for 
From the above, we have:
Differentiating both sides, we obtain:
This simplifies to:
We can now re-use the expression for and obtain:
Using the expression
for 
We have:
Using the product rule for differentiation and the chain rule for differentiation, we get:
Note from the expression that shows that
is even, so we can rewrite
as
, and we get:
We can re-use the expression and obtain:
Functional equations
Symmetry equation
The logistic function has the property that its graph
has symmetry about the point
. Explicitly, it satisfies the functional equation:
We can see this algebraically:
Multiply numerator and denominator by , and get:
Differential equation
As discussed in the #First derivative section, the logistic function satisfies the condition:
Therefore, is a solution to the autonomous differential equation:
The general solution to that equation is the function where
. The initial condition
at
pinpoints the logistic function uniquely.
Points and intervals of interest
Critical points
The function has no critical points. To see this, note that the derivative is:
Note that the numerator is never zero, nor is the denominator. Therefore, the function is always defined and nonzero.
Intervals of increase and decrease
The derivative:
is always positive. So the function is increasing on all of .
The asymptotic values are:
and:
In other words, the range of the function is the open interval , and it increases throughout its domain.
Points of inflection
The second derivative is:
We already noted that is always defined and nonzero, so the only way for
to be zero is if
< or
. This solves to:
This solves to , or
.
Thus, the second derivative is 0 at the point , i.e., with
and
.
Intervals of concave up and down
As above, we have:
We also noted that for all
. Therefore,
for
and
for
. Therefore,
is:
- concave up for
, i.e.,
- concave down for
, i.e.,
Symmetry
We discussed above a functional equation satisfied by :
From this, the following can be deduced:
- The graph of
has half-turn symmetry about the point
.
-
is an even function. Note that this can also be seen from the actual expression:
. But we don't need the actual expression to deduce that it is even -- the functional equation above gives evenness.
-
is an odd function. This can be directly deduced from
being even, but can also be verified from the actual expression:
.
Integration
First antiderivative
Direct computation
We have:
Computation in terms of functional equations for the logistic function
We have:
We also have that , so we get:
This can be rewritten as:
By the chain rule for differentiation, we get:
Thus:
This can be simplified and verified to be the same as the answer obtained by direct computation.