The exponential function models exponential growth. The output of the function at any given point is equal to the rate of change at that point. For real number input, the function conceptually returns Euler's number raised to the value of the input.
exp(x)
Name  Description 

x  A real number, complex number or matrix. 
The exponential function models exponential growth and has unique properties that make calculating calculustype questions easier. The function has two important properties. First, the value of the function at is and second, the output of the function at any given point is equal to the rate of change at that point. For example, here are some examples of the output of the function.
At each of the points , and , the rate of change of the function is equal to the output of the function at that point. This property is why the exponential function appears in many formulas and functions to define a family of exponential functions. In addition to exhibiting the properties of the exponent operator, the function gives the family of functions useful properties and the variables more meaningful values.
Note, whenever the math expression appears in an equation, the equation can be transformed to use the exponential function as . Also, the exponential function is the inverse of the natural logarithm function.
The exponential function often appears in the shorthand form . The constant is Euler’s Number and is defined as having the approximate value of . This shorthand suggestively defines the output of the exponential function to be the result of raised to the th power, which is a valid way to define and think about the function^{[1]}.
However, this site considers purely as shorthand for and instead defines the exponential function using the power series (shown below) for a number reasons. The numeric value is still very important in calculus and is the output of the exponential function when given the input . This relationship is shown below.
The exponential function is formally defined by the power series. This definition can be derived from the concept of compound interest^{[2]} or by using a Taylor Series^{[3]}. Shown below is the power series definition:
Using a power series to define the exponential function has advantages: the definition verifies all of the properties of the function^{[4]}, outlines a strategy for evaluating fractional exponents, provides a useful definition of the function from a computational perspective, and helps visualize what is happening for input other than Real Numbers.
Shown below are the properties of the exponential function. The power series definition, shown above, can be used to verify all of these properties ^{[4]}.
Property  Equation 

Zero Property  
Derivative Property  
Inverse Property  
Addition Property  
Subtraction Property  
Multiplication Property  
General Exponentiation 
Most of these properties parallel the properties of exponentiation, which highlights an important fact about the exponential function. While the exponential function appears in many formulas and functions, it does not necassarily have to be there. An exponential expression where a base, such as and , is raised to a power can be used to model the same behavior.
The short answer to why the exponential function appears so frequenty in formulas is the desire to perform calculus; the function makes calculating the rate of change and the integrals of exponential functions easier.
The exponential function appears in numerous math and physics formulas. For example, it appears in the formula for population growth, the normal distribution and Euler’s Formula. Notably, the applications of the function are over continuous intervals.
The formula for population growth is a straightforward application of the function. Given an initial population size and a growth rate constant , the formula returns the population size after some time has elapsed.
Expression  Description 

The population at time .  
The initial population, sometimes modeled as so the formula returns the percentage of growth.  
Shorthand for the exponential function.  
The relative growth rate constant. If , the population is exponentially growing, if the population stays the same, and if the population is decaying or decreasing.  
The time elapsed. 
Note, the population formula does not explicitly have to use the exponential function. For example, the same exponential growth curve can be defined in the form or as another exponential expression with different base. By using the exponential function, the formula inherits a bunch of useful properties that make performing calculus a lot easier.
For example, say we have modeled a popuplation with a relative growth rate constant of and for simplicity’s sake let’s say the initial population is . This gives a function, shown below, that returns the population at some time .
Because the population is modeled using the exponential function, we can easily calculate the rate of change of the population by taking the derivative with respect to time of the function. This is shown below.
The derivative is calculated by applying the derivative property from above and using the chain rule. This gives us the derivative below which models the change with respect to time of the population.
In addition to Real Number input, the exponential function also accepts complex numbers as input. For applications of complex numbers, the function models rotation and cyclic type patterns in the two dimensional plane referred to as the complex plane.
The exponential function appears in what is perhapsthe most notorious math formula: Euler’s Formula. The formula takes in angle an input and returns a complex number that represents a point on the unit circle in the complex plane that corresponds to the angle. The definition of Euler’s formula is shown below.
The output of Euler’s formula can be visualized as a point on the unit circle in the complex plane as shown in the figure below.
See Euler’s Formula page for why the exponential function appears and how it relates to the trigonometric functions sine and cosine.
The exponentiation operator is a binary operator. The base is an expression or number that is being raised to some exponent. The exponent expression is denoted using superscript text.
Returns the natural logarithm of the number x.
Euler's number is a naturally occurring number related to exponential growth and exponential decay.
The population growth formula models the exponential growth of a function. Note, this formula models unbounded population growth. For bounded growth, see logistic growth.
The normal distribution is a continuous probability distribution that appears naturally in applications of statistics and probability. The shape of the function forms a "bellcurve" which is symmetric around the mean and whose shape is described by the standard deviation.
Euler's Formula returns the point on the unit circle in the complex plane when given an angle.
A complex number is an extension of the real number line where in addition to the "real" part of the number there is a complex part of the number. The properties of complex numbers are useful in applied physics as they elegantly describe rotation.

Derive Definition of Exponential Function (Euler's Number) from Compound InterestWumbo (internal)

Derive Definition of Exponential Function (Power Series) from Compound InterestWumbo (internal)

Derive Definition of Exponential Function (Taylor Series)Wumbo (internal)

Verify Exponential Function PropertiesWumbo (internal)