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How are exponential functions used to model radioactive decay?

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A graph of the function f(x) = x cubed minus 3x plus 2 is shown on a blue background, with a red curve passing through key points. Local maximum and minimum points are marked with black dots at x equals negative 1 and x equals 1, illustrating important features of the curve. — function Plot of the cubic equation f(x) = x³ − 3x + 2. The plotted points are where changes in curvature occur.

Radioactive decay is modeled using exponential functions because the rate at which a substance decays is directly proportional to the amount of material currently present.12

This process is described by the differential equation $d N / d t = - λ N$ , where $N$ is the number of radioactive atoms at time $t$ , and $λ$ is the decay constant. Integrating this equation yields the exponential decay formula:34

$N = N_{0} e^{- λ t}$

In this model, $N_{0}$ represents the initial population of atoms at time $t = 0$ , and $e$ is the base of the natural logarithm (approximately 2.718). Because the rate of change depends on the current value, the substance decreases by the same proportion over any fixed period of time. This behavior is also characterized by the half-life ( $T_{1 / 2}$ ), which is the time required for half of the original population to decay, related to the decay constant by $T_{1 / 2} = 0.693 / λ$ .341