The exponential function is a mathematical function denoted by ${\displaystyle f(x)=\exp(x)}$ or ${\displaystyle e^{x}}$ (where the argument x is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of exponentiation (repeated multiplication), but modern definitions (there are several equivalent characterizations) allow it to be rigorously extended to all real arguments, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to opine that the exponential function is "the most important function in mathematics".

The exponential function satisfies the exponentiation identity

While other continuous nonzero functions ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ that satisfy the exponentiation identity are also known as exponential functions, the exponential function exp is the unique real-valued function of a real variable whose derivative is itself and whose value at 0 is 1; that is, ${\displaystyle \exp '(x)=\exp(x)}$ for all real x, and ${\displaystyle \exp(0)=1.}$ Thus, exp is sometimes called the natural exponential function to distinguish it from these other exponential functions, which are the functions of the form ${\displaystyle f(x)=ab^{x},}$ where the base b is a positive real number. The relation ${\displaystyle b^{x}=e^{x\ln b}}$ for positive b and real or complex x establishes a strong relationship between these functions, which explains this ambiguous terminology.

The real exponential function can also be defined as a power series. This power series definition is readily extended to complex arguments to allow the complex exponential function ${\displaystyle \exp :\mathbb {C} \to \mathbb {C} }$ to be defined. The complex exponential function takes on all complex values except for 0 and is closely related to the complex trigonometric functions, as shown by Euler's formula.

Motivated by more abstract properties and characterizations of the exponential function, the exponential can be generalized to and defined for entirely different kinds of mathematical objects (for example, a square matrix or a Lie algebra).

In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change (that is, percentage increase or decrease) in the dependent variable. This occurs widely in the natural and social sciences, as in a self-reproducing population, a fund accruing compound interest, or a growing body of manufacturing expertise. Thus, the exponential function also appears in a variety of contexts within physics, computer science, chemistry, engineering, mathematical biology, and economics.

The real exponential function is a bijection from ${\displaystyle \mathbb {R} }$ to ${\displaystyle (0;\infty )}$. Its inverse function is the natural logarithm, denoted ${\displaystyle \ln ,}$ ${\displaystyle \log ,}$ or ${\displaystyle \log _{e};}$ because of this, some old texts refer to the exponential function as the antilogarithm.