Exponential

An exponential refers to a function or phenomenon in which a quantity grows or decays at a rate proportional to its current value. In mathematics, an exponential function has the form f(x) = a b^x with a > 0 and b > 0, b ≠ 1. The most common case is the natural exponential function f(x) = e^x, where e is the base of the natural logarithm. Exponentials are invertible by the logarithm, with log_b(f(x)) = x for appropriate bases.

Exponential growth describes a quantity that increases by a constant multiple per unit time, modeled as y = y0 e^{k t} with k > 0. Exponential decay occurs when k < 0. Such processes double or halve at rates determined by k, with doubling time t_d = ln 2 / k for growth.

Key properties include that the derivative of e^{k x} is k e^{k x}, and more generally the derivative of a^x is a^x ln a. Exponentials are positive for all real x, infinitely differentiable, and their graphs are convex and strictly increasing (for base > 1) or decreasing (for 0 < base < 1).

Applications span science and finance. Continuous compounding in finance is modeled by A = P e^{r t}. In biology and chemistry, populations and concentrations often follow exponential laws, at least over limited ranges. In probability theory, the exponential distribution with density f(x) = λ e^{-λ x} for x ≥ 0 is memoryless.