Exponential Function

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.

exp(x)

The exponential function models exponential growth and has unique properties that make calculating calculus-type 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].

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 necessarily 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 frequently 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.

Note, that 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 a 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 population 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 perhaps the most notorious math formula: Euler’s Formula. The formula takes in an angle as 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.

1. Derive Definition of Exponential Function (Euler's Number) from Compound Interest Example
   https://wumbo.net/examples/derive-exponential-function-compound-interest-alt/
2. Derive Definition of Exponential Function (Power Series) from Compound Interest Example
   https://wumbo.net/examples/derive-exponential-function-compound-interest/
3. Derive Definition of Exponential Function (Taylor Series) Example
   https://wumbo.net/examples/derive-exponential-function-taylor-series/
4. Verify Exponential Function Properties Example
   https://wumbo.net/examples/verify-exponential-function-properties/