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Logistic Function Analyzer

Analyze f(x)=L/(1+e^{-k(x-x0)}) with carrying capacity and growth parameters

L= k= x0= Evaluate at x=

Logistic Function Properties

f(x)=L/(1+e^{-k(x-x0)})

As x->-inf: f->0, As x->+inf: f->L

Inflection at (x0, L/2), max growth rate

Symmetric about the inflection point

The logistic function models growth with a carrying capacity. It starts with exponential growth, then slows as it approaches the maximum. The inflection point marks the transition from accelerating to decelerating growth.

⚠ L must be positive. k>0 gives increasing S-curve. k<0 gives decreasing curve. x0 shifts the curve horizontally.

What Is the Logistic Function?

The logistic function is an S-shaped curve ranging from 0 to L. It features an inflection point at x0 where the value is L/2. The growth rate k determines how steeply the curve rises. It is widely used in population biology, machine learning, and statistics.

Asymptotes

Lower: f->0 as x->-inf. Upper: f->L as x->+inf. The function stays between 0 and L.

Growth Rate

Max growth rate = k*L/4 at the inflection point. Growth accelerates before x0, decelerates after.

Symmetry

Logistic function is symmetric: f(x0+t) - L/2 = L/2 - f(x0-t). Point symmetry about (x0, L/2).

Sigmoid

Standard sigmoid: L=1, k=1, x0=0 gives f(x)=1/(1+e^{-x}). Maps R to (0,1). Used as activation in neural networks.

Teaching Example: f(x)=100/(1+e^{-0.5x}). L=100, k=0.5, x0=0. At x=5: exp(-0.5*5)=exp(-2.5)=0.082, f(5)=100/1.082=92.4. Inflection at (0,50). Carrying capacity=100.

Applications

Population Machine Learning Epidemiology Economics Technology

Frequently Asked Questions

What is logistic function?▼

S-curve f(x)=L/(1+e^{-k(x-x0)}). Starts at 0, ends at L. Inflection at x0 where value=L/2.

Carrying capacity L?▼

Maximum value as x->inf. Upper horizontal asymptote. Represents maximum sustainable value.

Inflection point?▼

At x=x0, f=L/2. Maximum growth rate occurs here. Growth changes from accelerating to decelerating.

Logistic vs exponential?▼

Exponential grows without bound. Logistic has carrying capacity L. Logistic is bounded exponential growth.

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