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Unit 2: Exponential & Logarithmic Functions

The Logistic Model

Modeling bounded growth with the logistic function.

Advanced25 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

  • Logarithms as Inverses

Growth That Knows When to Slow Down

Exponential growth never stops growing — every model built so far climbs (or decays) forever. Before reading on: what kind of formula could grow like an exponential at first, but level off toward a fixed maximum instead of climbing without bound?

Definition — Logistic Function

A logistic function has the form f(x) = L / (1 + A·e^(−kx)), where L is the carrying capacity — the value f(x) approaches but never reaches — and k > 0 controls how quickly growth happens. A is determined by the initial condition.
f(x)=L1+Ae−kxf(x) = \frac{L}{1 + A e^{-kx}}

The carrying capacity isn't an assumption bolted on — it falls straight out of the formula's limiting behavior. As x → ∞ (with k > 0), the exponent −kx → −∞, so e^(−kx) → 0, and the denominator approaches 1 + 0 = 1: f(x) → L/1 = L. As x → −∞, e^(−kx) → ∞, the denominator grows without bound, and f(x) → 0. The function is squeezed between two horizontal asymptotes, y = 0 and y = L, by construction.

Worked Example — Building and Evaluating a Logistic Model

A population has carrying capacity L = 500, starts at f(0) = 50, and has growth parameter k = 0.4. Find A from the initial condition: 500/(1 + A) = 50, so 1 + A = 10 and A = 9. The model is f(x) = 500/(1 + 9e^(−0.4x)). Evaluate f(5): e^(−2) ≈ 0.13534, so 9 × 0.13534 ≈ 1.2180, giving f(5) = 500/2.2180 ≈ 225.4. After 5 units of x, the population has reached about 225 — 45% of the carrying capacity — and growth is beginning to slow as it approaches 500.

Worked Example — Finding Where Growth Is Fastest

A logistic curve is symmetric around the point where it crosses half its carrying capacity, L/2. Using the model above, solve 500/(1 + 9e^(−0.4x)) = 250: 1 + 9e^(−0.4x) = 2, so 9e^(−0.4x) = 1, e^(−0.4x) = 1/9, and −0.4x = −ln(9), giving x = ln(9)/0.4 ≈ 2.1972/0.4 ≈ 5.49. Near x ≈ 5.49, the population is climbing through its steepest, most exponential-looking stretch — before and after that point, the curve's growth visibly bends toward flat.

Function Explorer

Transform: g(x) = a·f(b(x − h)) + k
1
1
0
0
Composition
Analysis (of the transformed function, in view)
y-intercept
(0, 0)
x-intercepts
(-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Extrema
local min at (-7.85, -1); local max at (-4.71, 1); local min at (-1.57, -1); local max at (1.57, 1); local min at (4.71, -1); local max at (7.85, 1)
Inflection points
(-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Vertical asymptotes
none found in view
Horizontal asymptotes
none found
Domain
all real numbers in view
Range (estimated)
approximately [-1, 1]

Tip

Sanity-check any logistic model by confirming f(0) matches the stated initial condition and f(x) never actually reaches L — if a computed value equals or exceeds L, a sign or algebra error happened upstream.

Common Mistakes

  • Treating a logistic model as if it grows exponentially forever, the way f(x) = a · bˣ does.

    A logistic model only behaves like exponential growth near the start, while the denominator's e^(−kx) term still dominates — it bends and approaches the carrying capacity L as x grows, never exceeding it.

  • Forgetting the negative sign in the exponent when evaluating e^(−kx), and computing explosive growth instead of decay toward 0.

    The exponent is −kx, not kx — with k > 0 and x increasing, that exponent becomes more negative, sending e^(−kx) toward 0, not toward infinity.

Key Takeaways

  • A logistic function f(x) = L/(1 + A·e^(−kx)) has two horizontal asymptotes, y = 0 and y = L, that follow directly from e^(−kx)'s limiting behavior.
  • The parameter A is found from an initial condition; L is the value the model approaches but never reaches.
  • A logistic curve looks exponential near its start and flattens as it approaches its carrying capacity — bounded growth, not unbounded growth.

Summary

This closes Unit 2: exponential, logarithmic, and logistic functions all describe growth, but only logistic growth builds in its own ceiling. The next unit turns to a different kind of repeating structure entirely — periodic behavior, modeled with trigonometric and polar functions.

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