Exponential Function

Summary

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.

Syntax

exp(x)

Arguments

Name Description
x A real number, complex number or matrix.

Usage

The exponential function models exponential growth and has unique properties that make calculating calculus-type questions easier. For the latter, the function has two important properties. 1) The value of the function at is and 2) the output of the function at any given point is equal to the rate of change at that point. For example, here is some output of the function.

At each of the points , and , the rate of change or, equivelantly, “slope” 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 exponentiation, 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.

Notation and Euler’s Number

Exponential Function Transformation for Euler's Number

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, we are going to consider purely as shorthand for and instead define the exponential function using the power series (shown below) for a number reasons. This choice gives us a deeper understanding of the function, allows us to verify all of its properties and sets us up for success when we discuss other input to the function besides real numbers.

Formal Definition

The exponential function is defined by the power series shown below. This definition can be derived from the concept of compound interest[2] or by using a Taylor Series[3].

Using this 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[5], and helps visualize what is happening for input other than Real Numbers.

Properties of the Exponential Function

Shown below are the properties of the function. The formal definition, shown above, can be used to verify all of these properties[4].

EquationName
Zero Identity
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[6]. See footnotes for longer answer.

Applications of the Exponential Function

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. As a tool, the exponential function provides an elegant way to describe continously changing growth and decay.

Population Growth Formula

The formula for population growth, shown below, 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.
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 since the initial population.

Note, as mentioned above, this 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[6].

Growth Rate Constant

In practice, the growth rate constant is calculated from data. For example, say we have two population size measurements and taken at time and . It’s tempting to say that the growth rate is , since the population doubled in unit of time, however this linear way of thinking is a trap. Instead, let’s solve the formula for and calculate the growth rate constant[7].

Given the growth constant, the exponential growth curve is now fitted to our original data points as shown in the figure below.

Population Growth Variations

Other useful variations of this formula are: 1) The logistic growth formula which models bounded population growth. 2) Given a initial population and growth rate, how long does it take for the population to double? 2) How to calculate the half life of a population in decay.

Complex Numbers Input

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.

Euler’s Formula

The exponential function appears in what is perhaps one of the most famous math formulas: 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. This section introduces complex number input and Euler’s formula simultaneously. Note, the math here gets a little tricky because of how many areas of math come together. The definition of Euler’s formula is shown below.

Euler’s formula can be visualized as, when given an angle, returning a point on the unit circle in the complex plane. This is shown in the figure below.

The implications of this behavior allow for some easy-to-calculate and elegant formulations of trigonometric identities. See Euler's Formula for a full discussion of why the exponential function appears and how it relates to the trigonometric functions sine and cosine.

References

Properties of Exponents | Concept

There are six properties of the exponent operator: the zero property, identity property, negative property, product property, quotient property, and the power property.

Natural Logarithm | Function

Returns the natural logarithm of the number x.

Euler's Number | Concept

Euler's number is a naturally occurring number related to exponential growth and exponential decay.

Population Growth | Formula

The population growth formula models the exponential growth of a function. Note, this formula models unbounded population growth. For bounded growth, see logistic growth.

Normal Distribution | Concept

The normal distribution is a continuous probability distribution that appears naturally in applications of statistics and probability. The shape of the function forms a "bell-curve" which is symmetric around the mean and whose shape is described by the standard deviation.

Euler's Formula | Concept

Euler's Formula returns the point on the the unit circle in the complex plane when given an angle.

Complex Number | Concept

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.

Footnotes

  1. Derive Definition of Exponential Function (Euler's Number) from Compound Interest
  2. Derive Definition of Exponential Function (Power Series) from Compound Interest
  3. Derive Definition of Exponential Function (Power Series) using Taylor Series
  4. Verify Exponential Function Properties
  5. Implement the Exponential Function
  6. Why is Euler's Number a Natural Choice
  7. Calculate Growth Rate Coeffecient