1Cademy - Exponential Growth and Decay Formula

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Natural Exponential Function

Formula

Exponential Growth and Decay Formula

Exponential functions can model growth and decay over time. For an original amount, $A_0$ , that grows or decays at a continuous rate, $k$ , for a certain time, $t$ , the final amount, $A$ , is given by the formula: $A = A_0 e^{kt}$ . This formula has the same structure as the continuous compound interest formula, replacing the principal $P$ with the original amount $A_0$ . In this model, the sign of the constant determines the behavior: exponential growth corresponds to a positive rate of growth or growth constant ( $k > 0$ ), while exponential decay corresponds to a negative rate of growth or decay constant ( $k < 0$ ).