Definition

Exponential decay happens when something keeps shrinking by the same fraction each time period, decreasing quickly at first and then more and more slowly, never quite reaching zero. It follows $A(t) = A_0 \cdot b^t$ where $0 < b < 1$, or equivalently $A(t) = A_0 \cdot e^{-rt}$ where $r > 0$ is the decay rate, with half-life $T_{1/2} = \ln(2)/r$ the time to reduce to half. It solves the ODE $dA/dt = -rA$ ($r > 0$): $A(t) = A_0 e^{-rt}$; in quantum mechanics, unstable particles decay exponentially, the probability of surviving to time $t$ being $e^{-t/\tau}$ where $\tau$ is the mean lifetime, and radioactive decay follows this law exactly at the level of individual nuclei, probabilistically.