Exponential function

The exponential function is one of the most important functions in mathematics. The application of this function to a value x is written as exp(x). Equivalently, this can be written in the form ex, where e is a mathematical constant, the base of the natural logarithm, which equals approximately 2.718281828, and is also known as Euler's number.

As a function of the real variable x, the graph of y=ex is always positive (above the x axis) and increasing (viewed left-to-right). It never touches the x axis, although it gets arbitrarily close to it (thus, the x axis is a horizontal asymptote to the graph). Its inverse function, the natural logarithm, ln(x), is defined for all positive x.

Sometimes, especially in the sciences, the term exponential function is more generally used for functions of the form kax, where a, called the base, is any positive real number not equal to one. This article will focus initially on the exponential function with base e, Euler's number.

In general, the variable x can be any real or complex number, or even an entirely different kind of mathematical object; see the formal definition below.