Exponential Decay
Functions & Advanced AlgebraExponential decay occurs when a quantity decreases by a constant percentage rate over equal time intervals, shrinking rapidly at first then more slowly.
Formula
A(t) = A_0 \cdot b^t, 0 < b < 1
Definition
Exponential decay happens when something keeps shrinking by the same fraction each time period. It decreases quickly at first, then more and more slowly.
Example
A hot cup of coffee cools by losing $10\%$ of its remaining heat each minute. Start at $80$ degrees: $80, 72, 64.8, 58.3, \ldots$ Each minute it loses less actual heat than the minute before.
Key Insight
Exponential decay never quite reaches zero but gets closer and closer. It is the mathematical description of "gradually fading away" seen in radioactive decay, medicine in the body, and battery discharge.
Definition
Exponential decay follows $A(t) = A_0 \cdot b^t$ where $0 < b < 1$, or $A(t) = A_0 \cdot e^{-rt}$ where $r > 0$ is the decay rate. The quantity decreases by the same percentage each unit of time. The half-life $T_{1/2} = \ln(2)/r$ is the time to reduce to half.
Example
A drug in the bloodstream: initial dose $200$ mg, decays at $r = 0.1$ per hour. $A(t) = 200e^{-0.1t}$. After $5$ hours: $A(5) = 200e^{-0.5} \approx 121$ mg. Half-life: $\ln(2)/0.1 \approx 6.93$ hours.
Key Insight
The same percentage decrease per unit time is the hallmark of exponential decay. This is fundamentally different from linear decay (same amount lost each period). Real-world cooling, discharge, and pharmacokinetics follow this pattern.
Definition
Exponential decay solves $dA/dt = -rA$ ($r > 0$): $A(t) = A_0 e^{-rt}$. In quantum mechanics, unstable particles decay exponentially: the probability of surviving to time $t$ is $e^{-t/\tau}$, where $\tau$ is the mean lifetime. Radioactive decay follows this law exactly (at the level of individual nuclei, probabilistically).
Example
Carbon-14 decay: $A(t) = A_0 e^{-\lambda t}$, $\lambda = \ln(2)/5730$ yr$^{-1}$. Radiocarbon dating measures the ratio $A(t)/A_0$ to determine $t$. Requires knowing the initial ratio and assuming decay follows this law.
Key Insight
Exponential decay is memoryless: given a nucleus has survived to time t, the distribution of its remaining lifetime is identical to the original distribution. This is the continuous-time analog of the geometric distribution in discrete probability.