eksponencialit

Eksponencialit, also known as exponentiality, is the property of a process where the rate of change is proportional to the current value, producing exponential growth or decay. In continuous time, the size y(t) follows y(t) = y0 e^(kt), where k is the exponential rate. In discrete time, y_n = y0 (1 + r)^n, with r the per-period growth rate.

The exponential function arises from the constant-relative-growth assumption; its base e is the natural base, about 2.71828. A constant k yields a constant relative growth rate, so the proportional change per time unit is fixed.

Key measures include the doubling time T2 = ln(2)/k for k>0 and the half-life T1/2 = ln(2)/|k| for k<0. Common examples are compound interest A = P e^(rt), population growth under unrestricted conditions, and radioactive decay N(t) = N0 e^(-λ t). Early-stage epidemics are often approximated as exponential.

Eksponencialit is related to, but distinct from, exponential distributions that model waiting times in Poisson processes. Real systems may deviate due to resource limits, noise, or changing rates, leading to logistic growth or other nonlinear dynamics.

Historically, the concept developed through contributions by de Moivre, Euler, and others, formalized as the exponential function and its differential equation. Today, eksponencialit is a fundamental tool across science and engineering for modeling growth, decay, and compound processes.