# 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 , 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 |

antiderivative | the function |

critical points | none |

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 |

## 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.