Logistic Function: Equation, Graph & Examples

Logistic Function is a model of the exponential growth of the population. It is a part of an exponential function that also considers the carrying capacity of the land. 

• Logistic Function involves limiting the growth of the population.
• It is also known as the logistics curve.
• The function will form a S-shaped curve with respect to the equation.
• The initial curve will show geometric growth followed by a saturation point.
• Later, growth begins to slow arithmetically and stops when it reaches the saturation point.
• Neural networks and logistic regression are some common examples of logistic functions.
• Mathematically, it can be represented as:

Key Terms: Logistic function, Sigmoid function, Exponential function, Function, Differential equation, Saturation Point, Logarithm, Population, Differential Equation

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What is Logistic Function?

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Logistic Function is a form of model that explains the exponential growth of population which also take into consideration the carrying capacity of land.

• It is an exponential function that considers definite areas will not have unlimited growth.
• When one population grows, the available resources decrease.
• It consist of inflection point where curve changes from up or down to sloping down or up.
• These are also known as concave upward or concave downward curve.
• One of the important feature includes it is the output of sigmoid function.
• It is used in the field of statistics, ecology, medicine and agriculture.
• The concept was introduced by Pierre Francois Verhulst.

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Logistic Function Equation

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The logistic function can be derived in several ways and each function arrives at the same logic. Here, we will take into account parameters that include p=1, x₀ = 0 and m= 1.

Thus, the logistic function reduces as below:

\(f (x) = \frac{1}{1 + e^{-x}} = \frac{e^x}{e^x + 1} = \frac{1}{2} + \frac{1}{2} tanh (\frac{x}{2})\)

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Logistic Function Curve

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The logistic function also known as logistic curve will form a S-shaped curve. The curve will first increase exponentially then it will reach its saturation point. After that curve begins to decrease linearly.

• The growth will stop when the curve will reach its maturation point.
• The Logistic Curve is also known as the Sigmoid curve because of its ‘S-shaped curve.
• The exponential or logistic function in the denominator determines the speed at which a function drops or grows to its limiting value.

f (x) = \(\frac{m} {1 + e^{ – p (x – x_0)}}\)

Where,

• M = The value that is maximum of the curve
• e = the natural logarithm base 
• x₀ = the given value of the midpoint of sigmoid
• P = The logistic growth rate or the curve’s steepness 

Logistic Curve

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Interpretation of Logistic Function

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The logistic function can be interpreted as follows:

• Consider, k (t) = the population as a function of time in a number
• t₀ = the starting time.
• The term (t - t_o) is just an adjustable horizontal translation of the logistic function.
• M = criterion that influences the rate of exponential growth
• D= the asymptote in horizontal or the limit on the population size
• The functions stand as

k(t) = \(\frac{D}{ (1+e^{-m(t-t_0))}}\)

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Derivative of Logistic Function

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The derivative of logistic function can be explained as follows:

Part 1

The derivation of logistic function are as follows:

f(x) = \(\frac{1}{1+e^{-x}} = \frac{e^x}{1+e^x}\) 

\(\frac{d}{d_x}\)f(x) = \(\frac{ex. (1+e^x) - e^x. e^x}{(1+e^x)^2} = \frac{e^x}{(1+e^x)^2}\) = f(x) (1 – f (x) )

• The above derivation is also known as logistic distribution.

Part 2

The logistic function is also derived from the differential equation. In this derivation, the logistic model states that the growth decreases linearly when the population increases.

• The functions are as given below:

\(\frac{dm (t)}{dt}\) = m (t) k [1 – \(\frac{m(t)}{B}\)]

Where,

• K > 0,
• B is a constant that is greater than the value of m(0).
• The equation shows the saturation of the population.
• It also expresses how the growth rate is decreasing as m(t) as it approaches B. 
• However, there is a solution to the above equation:

m (t) = \(\frac{B}{1+(B/m(0)-1)e^{kt}}\)

• The above function grows as an exponential function.
• It then falls and converges to carrying capacity B which is the maximum sustainable population.

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Integrals of the Logistic Function

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Consider the equation: 1- e^x= p

• After applying integration function the equation is as follows:

f (x) = \(\frac{e^x}{1+e^x} = \frac{\hat{p}}{p}\)

\(\int \frac{e^x}{1+e^x}\) dx= \(\int \frac{1}{p}\) dp = log p = log (1 + e^x)

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Logistic Function vs Sigmoid Function

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The basic difference between Logistic Function vs Sigmoid Function is the expression or formula that clearly differentiates between the two.

• The sigmoid function is an expression of a mathematical function which is S-shaped known as the sigmoid curve.
• The logistic function is the standard choice added for a sigmoid function.
• The return value of a sigmoid function is increasing from 0 to 1.
• It also include possible values from -1 to 1 and depends on convention.
• The value depends upon the kingdom for all real numbers. 
• The formula for Logistic function is given as:

f(x)= \(\frac{L}{1+e^{-k(x-x_0)}}\)

• The formula for Sigmoid Function is given as:

S(t)= \(\frac{1}{1+e^{-t}}\)

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Things to Remember

• The logistic function is an exponential function which considers the carrying capacity of the land.
• It proves that when growth increases, the population tends to decrease.
• The growth rate declines linearly when the population increases.
• The Logistic Curve is also known as the Sigmoid curve because of its ‘S-shaped curve.
• The function is used at the time of the pandemic.
• It determines the cumulative number of infected cases.