Exponential Growth
Exponential growth occurs when a quantity increases by a constant percentage rate over equal time intervals, causing it to grow faster and faster.
Formula
A(t) = A_0 \cdot b^t, b > 1
Definition
Exponential growth happens when a quantity keeps multiplying by the same number over and over: it starts slowly but then shoots up very fast. The model is $A(t) = A_0 \cdot b^t$, where $A_0$ is the initial amount, $b > 1$ is the growth factor, and $t$ is time, equivalently $A(t) = A_0 \cdot e^{rt}$ where $r > 0$ is the continuous growth rate. It is the solution to the ODE $dA/dt = rA$ ($r > 0$): $A(t) = A_0 e^{rt}$, the unique function (up to scaling) proportional to its own derivative; in discrete time it arises in linear maps with eigenvalue greater than $1$, and in population models as the Malthusian growth model.
Example
A bacteria population that doubles every hour follows $1, 2, 4, 8, 16, 32, \ldots$, reaching $1{,}024$ after $10$ hours and over a million after $20$, the growth accelerating rapidly. A $\$1{,}000$ investment at $6\%$ annual interest compounded annually, $A(t) = 1000 \cdot (1.06)^t$, grows to about $\$2{,}012$ after $12$ years, matching the "Rule of 72" estimate ($72/6 = 12$ years to double). The Lotka-Volterra predator-prey model departs from pure exponential growth by introducing nonlinear coupling: near equilibrium, linearization shows exponential modes, but far from equilibrium nonlinear terms dominate.
Key Insight
Exponential growth is deceptive because it looks slow at first and then becomes explosive, the first doubling doesn't seem like much, but eventually it takes over, and it has a constant doubling time, $\ln(2)/r$ regardless of when you start measuring, which is a defining property that distinguishes it from polynomial growth. Exponential growth is ultimately limited in physical systems: the logistic growth model $A'(t) = rA(1 - A/K)$ introduces a carrying capacity $K$, causing growth to slow and stabilize, approximating exponential behavior near $t=0$ but stabilizing near $K$.