Exponentials

Exponentials are functions defined by f(x) = a^x, where the base a is a positive real number not equal to 1. They are characterized by constant proportional change: f(x+1) = a f(x). If a > 1 the function grows with x; if 0 < a < 1 it decays. The natural exponential function, e^x, uses base e (approximately 2.71828) and plays a central role in analysis because it is its own derivative.

Key algebraic rules include a^{x+y} = a^x a^y, (a^x)^k = a^{kx}, and a^x > 0 for all real x. Exponentials are invertible with the logarithm: log_a(y) is the inverse of a^x. The calculus of exponentials satisfies d/dx a^x = a^x ln a, and ∫ a^x dx = a^x / ln a + C. When a = e, these simplify to d/dx e^x = e^x and ∫ e^x dx = e^x + C.

In many contexts the exponential function models processes with constant proportional change, such as population growth, compound interest, and radioactive decay. The natural exponential arises as the unique solution to the differential equation y' = y with y(0) = 1, and it underpins the exponential distribution in probability theory.

Complex exponentials extend the concept to the complex plane, with e^{iθ} = cos θ + i sin θ, linking exponentials to trigonometry. The exponential function thus occupies a foundational role across algebra, calculus, and applied disciplines.