- 1 Definition
- 2 Key data
- 3 Differentiation
- 4 Functional equations
- 5 Points and intervals of interest
- 6 Symmetry
- 7 Integration
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 .
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.
|default domain||all of , i.e., all reals|
|range||the open interval , i.e., the set|
|derivative|| the derivative is .|
If we denote the logistic function by the letter , then we can also write the derivative as
|second derivative||If we denote the logistic function by the letter , then we can also write the derivative as|
|logarithmic derivative|| the logarithmic derivative is |
If we denote the logistic function by , the logarithmic derivative is
|critical points for the derivative (correspond to points of inflection for the function)||; the corresponding point on the graph of the function is .|
|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 corresponding to the limit for |
asymptote at corresponding to the limit for
|inverse function||inverse logistic function or log-odds function given by|
Consider the expression for :
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:
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
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:
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:
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
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
is always positive. So the function is increasing on all of .
The asymptotic values are:
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.,
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: .
Computation in terms of functional equations for the logistic function
We also have that , so we get:
This can be rewritten as:
By the chain rule for differentiation, we get:
This can be simplified and verified to be the same as the answer obtained by direct computation.