June 03, 2022

Exponential Functions - Formula, Properties, Graph, Rules

What’s an Exponential Function?

An exponential function measures an exponential decrease or increase in a certain base. For instance, let us assume a country's population doubles every year. This population growth can be depicted as an exponential function.

Exponential functions have numerous real-world uses. Expressed mathematically, an exponential function is shown as f(x) = b^x.

In this piece, we discuss the basics of an exponential function along with relevant examples.

What is the equation for an Exponential Function?

The common equation 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 fixed, and x is a variable

As an illustration, 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 cases where b is higher than 0 and unequal to 1, x will be a real number.

How do you chart Exponential Functions?

To chart an exponential function, we need to locate the dots where the function intersects the axes. This is called the x and y-intercepts.

As the exponential function has a constant, it will be necessary to set the value for it. Let's focus on the value of b = 2.

To locate the y-coordinates, one must to set the worth for x. For instance, for x = 2, y will be 4, for x = 1, y will be 2

In following this approach, we get the domain and the range values for the function. Once we have the values, we need to plot them on the x-axis and the y-axis.

What are the properties of Exponential Functions?

All exponential functions share comparable properties. When the base of an exponential function is greater than 1, the graph would have the following characteristics:

  • The line intersects the point (0,1)

  • The domain is all positive real numbers

  • The range is more than 0

  • The graph is a curved line

  • The graph is increasing

  • The graph is smooth and continuous

  • As x advances toward negative infinity, the graph is asymptomatic towards the x-axis

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

In instances where the bases are fractions or decimals between 0 and 1, an exponential function exhibits the following properties:

  • The graph intersects the point (0,1)

  • The range is more than 0

  • The domain is all real numbers

  • The graph is decreasing

  • The graph is a curved line

  • As x nears positive infinity, the line within graph is asymptotic to the x-axis.

  • As x gets closer to negative infinity, the line approaches without bound

  • The graph is smooth

  • The graph is unending


There are several essential rules to bear in mind when engaging 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 with a base of 2, then we can note it as 2^x * 2^y = 2^(x+y).

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

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

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

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

Rule 4: An exponential function that has 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 identical to 0.

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


Exponential functions are generally leveraged to indicate exponential growth. As the variable increases, the value of the function rises faster and faster.

Example 1

Let’s observe the example of the growth of bacteria. Let us suppose that we have a group of bacteria that doubles hourly, then at the end of hour one, we will have 2 times as many bacteria.

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

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

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

f(t) = 2^t

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

Example 2

Moreover, exponential functions can represent exponential decay. If we have a dangerous substance that degenerates at a rate of half its quantity every hour, then at the end of the first hour, we will have half as much substance.

After two hours, we will have 1/4 as much material (1/2 x 1/2).

At the end of hour three, we will have one-eighth as much substance (1/2 x 1/2 x 1/2).

This can be represented using an exponential equation as follows:

f(t) = 1/2^t

where f(t) is the volume of material at time t and t is assessed in hours.

As demonstrated, both of these samples follow a similar pattern, which is the reason they can be represented using exponential functions.

As a matter of fact, any rate of change can be denoted using exponential functions. Bear in mind that in exponential functions, the positive or the negative exponent is depicted by the variable while the base stays fixed. Therefore any exponential growth or decline where the base changes is not an exponential function.

For instance, in the case of compound interest, the interest rate remains the same while the base changes in normal intervals of time.


An exponential function is able to be graphed employing a table of values. To get the graph of an exponential function, we must enter different values for x and measure the corresponding values for y.

Let's 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 demonstrated, the worth of y rise very quickly as x increases. If we were to draw this exponential function graph on a coordinate plane, it would look like this:

As seen above, the graph is a curved line that goes up from left to right ,getting steeper as it persists.

Example 2

Graph the following exponential function:

y = 1/2^x

To start, let's draw up a table of values.

As you can see, the values of y decrease very rapidly as x rises. The reason is because 1/2 is less than 1.

If we were to graph the x-values and y-values on a coordinate plane, it would look like this:

This 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 written as f(ax)/dx = ax. All derivatives of exponential functions present unique properties where 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 figure. The common form of an exponential series is:


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