June 03, 2022

Exponential Functions - Formula, Properties, Graph, Rules

What’s an Exponential Function?

An exponential function calculates an exponential decrease or rise in a particular base. For example, let us assume a country's population doubles yearly. This population growth can be represented in the form of an exponential function.

Exponential functions have multiple real-world uses. Mathematically speaking, an exponential function is shown as f(x) = b^x.

Here we will learn the fundamentals of an exponential function in conjunction with relevant examples.

What is the formula for an Exponential Function?

The generic formula for an exponential function is f(x) = b^x, where:

  1. b is the base, and x is the exponent or power.

  2. b is a constant, and x varies

For instance, if b = 2, we then get the square function f(x) = 2^x. And if b = 1/2, then we get the square function f(x) = (1/2)^x.

In a situation where b is larger than 0 and unequal to 1, x will be a real number.

How do you chart Exponential Functions?

To plot an exponential function, we need to find the spots where the function intersects the axes. This is referred to as the x and y-intercepts.

Considering the fact that the exponential function has a constant, one must set the value for it. Let's take the value of b = 2.

To locate the y-coordinates, its essential to set the rate for x. For example, for x = 1, y will be 2, for x = 2, y will be 4.

According to this technique, we get the range values and the domain for the function. Once we determine the rate, we need to plot them on the x-axis and the y-axis.

What are the properties of Exponential Functions?

All exponential functions share identical qualities. When the base of an exponential function is greater than 1, the graph is going to have the below characteristics:

  • The line passes the point (0,1)

  • The domain is all positive real numbers

  • The range is larger than 0

  • The graph is a curved line

  • The graph is rising

  • The graph is level and constant

  • As x nears negative infinity, the graph is asymptomatic regarding the x-axis

  • As x advances toward positive infinity, the graph rises without bound.

In cases where the bases are fractions or decimals within 0 and 1, an exponential function presents with the following qualities:

  • The graph intersects the point (0,1)

  • The range is larger than 0

  • The domain is all real numbers

  • The graph is decreasing

  • The graph is a curved line

  • As x advances toward positive infinity, the line in the graph is asymptotic to the x-axis.

  • As x advances toward negative infinity, the line approaches without bound

  • The graph is level

  • The graph is continuous


There are several basic rules to remember when dealing with exponential functions.

Rule 1: Multiply exponential functions with an identical base, add the exponents.

For instance, if we have to multiply two exponential functions that posses a base of 2, then we can compose it as 2^x * 2^y = 2^(x+y).

Rule 2: To divide exponential functions with the same base, deduct the exponents.

For example, if we need to divide two exponential functions that have a base of 3, we can compose it as 3^x / 3^y = 3^(x-y).

Rule 3: To raise an exponential function to a power, multiply the exponents.

For example, if we have to raise an exponential function with a base of 4 to the third power, then we can write it as (4^x)^3 = 4^(3x).

Rule 4: An exponential function with a base of 1 is always equal to 1.

For example, 1^x = 1 no matter what the rate of x is.

Rule 5: An exponential function with a base of 0 is always equal to 0.

For example, 0^x = 0 despite whatever the value of x is.


Exponential functions are usually used to signify exponential growth. As the variable rises, the value of the function grows at a ever-increasing pace.

Example 1

Let’s examine the example of the growth of bacteria. Let us suppose that we have a cluster of bacteria that multiples by two every hour, then at the close of hour one, we will have double as many bacteria.

At the end of the second hour, we will have 4x as many bacteria (2 x 2).

At the end of hour three, we will have 8x as many bacteria (2 x 2 x 2).

This rate of growth can be displayed an exponential function as follows:

f(t) = 2^t

where f(t) is the amount of bacteria at time t and t is measured in hours.

Example 2

Also, exponential functions can portray exponential decay. If we have a radioactive material that decomposes at a rate of half its quantity every hour, then at the end of one hour, we will have half as much substance.

At the end of two hours, we will have 1/4 as much substance (1/2 x 1/2).

At the end of the third hour, we will have 1/8 as much material (1/2 x 1/2 x 1/2).

This can be represented using an exponential equation as below:

f(t) = 1/2^t

where f(t) is the amount of material at time t and t is calculated in hours.

As shown, both of these samples use a similar pattern, which is the reason they are able to be shown using exponential functions.

In fact, any rate of change can be demonstrated using exponential functions. Recall that in exponential functions, the positive or the negative exponent is represented by the variable while the base remains constant. This means that any exponential growth or decay where the base is different is not an exponential function.

For instance, in the matter of compound interest, the interest rate continues to be the same while the base varies in normal intervals of time.


An exponential function can be graphed utilizing a table of values. To get the graph of an exponential function, we have to enter different values for x and then calculate the equivalent values for y.

Let us look at the following example.

Example 1

Graph the this exponential function formula:

y = 3^x

First, let's make a table of values.

As shown, the rates of y rise very rapidly as x increases. Imagine we were to draw this exponential function graph on a coordinate plane, it would look like the following:

As shown, the graph is a curved line that rises from left to right ,getting steeper as it goes.

Example 2

Chart the following exponential function:

y = 1/2^x

First, let's create a table of values.

As you can see, the values of y decrease very quickly as x increases. This is because 1/2 is less than 1.

If we were to chart the x-values and y-values on a coordinate plane, it is going to look like what you see below:

The above is a decay function. As you can see, the graph is a curved line that decreases from right to left and gets flatter as it proceeds.

The Derivative of Exponential Functions

The derivative of an exponential function f(x) = a^x can be displayed as f(ax)/dx = ax. All derivatives of exponential functions display particular properties by which the derivative of the function is the function itself.

The above can be written as following: f'x = a^x = f(x).

Exponential Series

The exponential series is a power series whose expressions are the powers of an independent variable number. The general form of an exponential series is:


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