exponential function

exponential function, in mathematics, a relation of the form y = a^x, where a is a fixed positive real number not equal to 1 and x is a real variable (the exponent). The domain of y = a is all real numbers, and its range is (0, ∞). The function is increasing when a > 1 and decreasing when 0 < a < 1.

An important exponential function is the natural exponential function y = e^x, also written exp ⁡(x),where the value of e is approximately 2.7182818, the base of the natural logarithm. The exponential and logarithmic functions are inverses: If y = a^x, then x = log⁡_ay; in particular, if y = e^x, then x = lny.

The exponential function is also defined as the sum of the infinite series:$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+\cdots$,which converges—meaning the sum approaches a single finite value—for all x. Here n! (n factorial) is the product of the first n positive integers, with 0! = 1. Setting x = 1 gives a series for the constant e:$e^1=1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\cdots \approx 2.7182818$.

A property of the exponential function is that its derivative equals the function itself:$\frac{d}{dx}{e}^x=e^x$.More generally,$\frac{d}{dx}{a}^x=a^x\ln a$.

The exponential functions are examples of nonalgebraic, or transcendental, functions—i.e., functions that cannot be represented as the product, sum, and difference of variables raised to some nonnegative integer power. Other common transcendental functions are the logarithmic functions and the trigonometric functions. Exponential functions frequently arise and quantitatively describe a number of phenomena in physics, such as radioactive decay, in which the rate of change in a process or substance depends directly on its current value.