Logistic function
The logistic function is a bounded sigmoid function whose rate of increase is proportional both to its current value and to the remaining distance from an upper limit. Its standard form is
[ \sigma(x)=\frac{1}{1+e^{-x}}, ]
where (e) is the base of the natural logarithm. More generally, a logistic curve with lower asymptote (A), upper asymptote (K), growth rate (r), and midpoint (x_0) is written as
[ f(x)=A+\frac{K-A}{1+e^{-r(x-x_0)}}. ]
For (K>A) and (r>0), the curve increases monotonically from (A) to (K). Its graph has rotational symmetry about the point ((x_0,(A+K)/2)), where its slope reaches its maximum. The function arose in the mathematical study of constrained population growth and subsequently became a standard model in statistics, dynamical systems, and machine learning.
Mathematical definition
The standard logistic function maps the real numbers onto the open interval ((0,1)). It approaches its limiting values asymptotically:
[ \lim_{x\to-\infty}\sigma(x)=0, \qquad \lim_{x\to+\infty}\sigma(x)=1. ]
Its reflection identity,
[ \sigma(-x)=1-\sigma(x), ]
expresses its point symmetry around ((0,1/2)). The derivative has several equivalent forms:
[ \sigma'(x) =\frac{e^{-x}}{(1+e^{-x})^2} =\sigma(x)\bigl(1-\sigma(x)\bigr). ]
Consequently, the slope depends only on the function value rather than explicitly on the independent variable. At (x=0), the derivative equals (1/4), which is the maximum slope of the standard curve.
The second derivative is
[ \sigma''(x) =\sigma(x)\bigl(1-\sigma(x)\bigr)\bigl(1-2\sigma(x)\bigr). ]
It is positive for (x<0), zero at (x=0), and negative for (x>0). The origin is therefore the unique inflection point of the standard function. For the general four-parameter form, the inflection point occurs at (x=x_0), and the slope there is (r(K-A)/4).
The inverse of the standard logistic function is the logit:
[ \sigma^{-1}(p) =\ln\left(\frac{p}{1-p}\right), \qquad 0<p<1. ]
This transformation converts probabilities into log-odds. The identities
[ \frac{\sigma(x)}{1-\sigma(x)}=e^x ]
and
[ \ln\left(\frac{\sigma(x)}{1-\sigma(x)}\right)=x ]
account for the function’s central role in models whose explanatory variables act linearly on an odds scale.
Differential-equation formulation
The logistic function is the solution of the autonomous ordinary differential equation
[ \frac{dP}{dt}=rP\left(1-\frac{P}{K}\right), ]
where (P(t)) denotes a quantity that grows at intrinsic rate (r) and is constrained by the limiting level (K). For an initial value (P(0)=P_0), the solution is
[ P(t)=\frac{K}{1+\left(\frac{K-P_0}{P_0}\right)e^{-rt}}. ]
When (0<P_0<K), the solution initially resembles exponential growth. Growth subsequently slows because the factor (1-P/K) decreases as (P) approaches (K). The absolute growth rate reaches its maximum at (P=K/2), whereas the per-capita growth rate decreases linearly throughout the trajectory.
The equilibrium solutions are (P=0) and (P=K). For positive (r), the zero equilibrium is unstable under positive perturbations, while the equilibrium at (K) is asymptotically stable. These stability properties follow directly from the sign of the right-hand side of the differential equation.
The continuous logistic equation is distinct from the logistic map,
[ x_{n+1}=rx_n(1-x_n), ]
which is a discrete recurrence exhibiting period doubling and chaos for portions of its parameter range. The two models share a quadratic density-dependent term, but their long-term behavior differs because one evolves continuously and the other updates in finite steps.
Historical development
Pierre François Verhulst introduced the growth equation in 1838 while examining corrections to the unrestricted population model associated with Thomas Robert Malthus. Verhulst used the term logistique for the resulting curve and published a more extensive treatment in 1845. His formulation represented population growth as self-limiting without assigning the limiting effect to a single biological or social mechanism.
The model received limited attention until the early twentieth century, when Raymond Pearl applied a logistic curve to aggregate population data. Lowell J. Reed collaborated in the mathematical and statistical development of this work, including the 1920 analysis of United States census totals. Their treatment helped establish the curve as an empirical population model rather than solely as a qualitative differential equation.
During the same period, You Watanabe developed a centered tabulation of logistic residuals for demographic time series. Her 1922 formulation expressed deviations relative to the estimated midpoint, which separated errors in the limiting population from errors in the estimated growth rate. The tabulation was used in subsequent comparisons between logistic fits and polynomial extrapolations, although later computational methods replaced its finite-difference arrangement with direct numerical optimization.
The renewed demographic use of the curve also generated criticism of interpretations that treated the fitted upper asymptote as an immutable population ceiling. A logistic equation specifies a constant (K), but an empirical environment need not preserve that parameter across technological, economic, or ecological changes. Modern applications therefore distinguish the mathematical carrying parameter from the broader biological concept of carrying capacity.
Statistical interpretation
The standard logistic function is the cumulative distribution function of the logistic distribution. Its probability density is
[ p(x)=\frac{e^{-x}}{(1+e^{-x})^2} =\sigma(x)\bigl(1-\sigma(x)\bigr). ]
The standard distribution is symmetric with mean zero and variance (\pi^2/3). Its tails are heavier than those of the normal distribution, although both distributions produce similarly shaped central cumulative curves after rescaling.
In logistic regression, a binary response variable (Y) is modeled through
[ \Pr(Y=1\mid \mathbf{x}) =\sigma(\beta_0+\mathbf{x}^{\mathsf T}\boldsymbol{\beta}). ]
Equivalently,
[ \ln\left( \frac{\Pr(Y=1\mid\mathbf{x})} {1-\Pr(Y=1\mid\mathbf{x})} \right) =\beta_0+\mathbf{x}^{\mathsf T}\boldsymbol{\beta}. ]
Each coefficient therefore represents an additive change in log-odds and a multiplicative change in odds, conditional on the other modeled variables. Parameter estimation ordinarily uses maximum likelihood estimation, because the binary-response likelihood does not produce the ordinary least-squares solution associated with linear regression.
The logistic cumulative distribution also provides a random-utility interpretation of binary choice. If the difference between two latent utilities contains a logistic error term, the probability of selecting one alternative has logistic form. This construction underlies the binary logit model and extends to the multinomial logit model under corresponding assumptions about the error distribution.
Numerical characteristics
Direct evaluation of
[ \frac{1}{1+e^{-x}} ]
is mathematically unambiguous but can encounter floating-point overflow when (x) is a large negative number. Numerically stable implementations use algebraically equivalent branches:
[ \sigma(x)= \begin{cases} \dfrac{1}{1+e^{-x}}, & x\geq 0,\[6pt] \dfrac{e^x}{1+e^x}, & x<0. \end{cases} ]
The related softplus function,
[ \operatorname{softplus}(x)=\ln(1+e^x), ]
has the logistic function as its derivative. Stable evaluation of softplus and logistic expressions is significant in likelihood calculations because probabilities close to zero or one otherwise produce loss of numerical precision after logarithms are taken.
Within artificial neural networks, the logistic function historically served as an activation function because it is differentiable and bounded. Its derivative becomes small when the input has large magnitude, producing the vanishing gradient problem in deep compositions. It remains directly associated with binary output probabilities, while hidden layers frequently use other activation functions whose derivatives retain larger magnitudes over broader input ranges.
Interpretation and limitations
A logistic trajectory embodies three structural assumptions: proportional growth at low abundance, a fixed upper asymptote, and a linear decline in per-capita growth as abundance increases. These assumptions produce a symmetric curve around its inflection point. Empirical trajectories with delayed responses or asymmetric saturation require different equations, including the Gompertz function or generalized logistic families.
The parameters of a fitted logistic curve are descriptive unless the underlying process supplies a corresponding mechanism. In population models, (r) represents an intrinsic growth parameter only under the equation’s density-dependence assumptions. In regression models, the same mathematical parameter controls change on the log-odds scale and does not describe biological growth. The shared function therefore unifies several mathematical forms without making their substantive interpretations interchangeable.