In this explainer, we will learn how to identify, write, evaluate, and analyze exponential functions.

### Definition: Exponential Function

An exponential function is a function with a rule of the form where the constant is the base and the variable is the exponent. The constant is a real number such that and , and the variable can be any real number.

To analyze an exponential function with the rule , let’s begin by finding for some positive integer values of .

1 | 2 | 3 | 4 | |

We can see that the previous value of is multiplied by every time increases by 1. That is,

- the value of is ,
- the value of is ,
- the value of is .

### Property: Relationship between 𝑓(𝑥)
and
𝑓(𝑥
−
1)
for the Exponential Function
𝑓(𝑥)
=
𝑏^{𝑥}

For the exponential function , the value of is always the product of and , which means that is always the quotient of and . That is,

The above property is true not only for positive integer values of , but for any real number value of . Thus, we can often use the relationship between , , and to find the value of from a graph or a table.

Consider the following graph, which shows for a specific value of . Notice that the graph intersects the -axis at 1. This will be the case for any real number value of such that and .

We can find the value of in the function that the graph represents by selecting two points that the graph passes through with -coordinates that differ by 1. It does not matter which points we choose, but both should have - and -coordinates that are easy to identify. In this case, it is easiest for us to select points with -coordinates that are consecutive integers.

It is clear that the graph passes through the points and , so we will choose these two points to find the value of . Based on the coordinates of the points, we know that

Remember that is always equal to the quotient of and . Therefore,

So, the graph represents the function .

Next, let’s look at a table representing the function for a specific value of .

1 | 2 | 3 | 4 | |

2 | 4 | 8 | 16 |

Since the values of in the table differ by 1, we could use the same procedure as above to determine the value of , finding either the quotient of 4 and 2, the quotient of 8 and 4, or the quotient of 16 and 8. However, this time, we will instead substitute the first pair of - and -values in the table into the rule . Substituting 1 for and 2 for gives us which tells us that the value of must be 2. Thus, the table represents the function .

So far, we have looked at only exponential functions in the form , but an exponential function can be transformed in the usual manner. While the value of the base must always be a positive real number that is not equal to 1, the exponent can consist of any number of linear terms, as long as at least one of them contains a variable. Also, sometimes the exponential term is multiplied by a constant, or a constant is added to it. Some examples of transformed exponential functions include:

- ,
- ,
- .

Just as with exponential functions in the form , we can see that in each of these examples, the exponent contains a variable, is raised, and is to the right of the base:

- for , the base is 2, and the exponent is ,
- for , the base is , and the exponent is ,
- for , the base is 9, and the exponent is .

We can also find the equations of transformed exponential functions such as these from graphs and tables, which we will do in some of the problems that follow. We will begin, though, with a problem that asks us to identify an exponential function’s base and exponent.

### Example 1: Identifying the Base and the Exponent of an Exponential Function

What are the base and exponent of the function ?

### Answer

Remember that for an exponential function with the rule , the base is the constant , and the exponent is the variable . We know that is a real number such that and and that can be any real number.

Our exponential function has been transformed. In it, we have the variable binomial expression raised and to the right of 5. Therefore, we know that the base is 5 and that the exponent is .

In the next problem, we are given a table of values for an exponential function and are asked to find the function’s equation.

### Example 2: Writing an Exponential Equation from a Table of Values

Write an exponential equation in the form for the numbers in the table.

2 | 4 | 5 | |

### Answer

Recall that for an exponential equation in the form , the base is the constant , and the exponent is the variable . We know that is a real number such that and and that can be any real number.

Here, since we are asked to write an exponential equation in the form for the numbers in the table, we must find the value of the base, , to solve the problem.

Let’s begin by substituting one of the pairs of - and -values from the table into the equation .

Substituting 2 for and for gives us

Now, to solve for , we can take the square root of both sides of the equation. A square root can be positive or negative, but since must be a positive real number, we can ignore the negative root, giving us

Remember that to find the square root of a fraction, we take the square root of both the numerator and the denominator, so our equation can be simplified as follows:

Thus, since the value of is , an exponential equation in the form for the numbers in the table is .

### Check

We can verify our answer by using another method to solve for the base, . To use this method, we will substitute two other pairs of - and -values from the table into the equation . First, we will substitute 4 for and for to get

Next, we will substitute 5 for and for to get

Now we can divide the second equation by the first equation. Doing so gives us

This equation can be rewritten as and simplified to

This confirms that the value of is and that an exponential equation in the form for the numbers in the table is .

Note that in the process of checking our answer, we used the
**quotient rule**, which is one of the properties of exponents.

### Definition: Quotient Rule

The **quotient rule** states that when dividing exponential expressions with the same base, we keep the base and find the difference of the exponents. That is,
where is the base and and
are the exponents.

For example, we would use the
**quotient rule** to find that
.

In the problem that follows, we are again given a table of values for an exponential function and are asked to find the function’s equation. This time, however, the equation is in a form other than .

### Example 3: Writing an Exponential Equation from a Table of Values

Write an exponential equation in the form for the numbers in the table.

0 | 1 | 2 | 3 | |

18 | 6 | 2 |

### Answer

In this problem, we are asked to write an exponential equation in the form for the numbers in the table, so we must find the values of and to solve the problem. Let’s begin by substituting one of the pairs of - and -values from the table into the equation .

Substituting 0 for and 18 for gives us

Since any base raised to the power 0 is equal to 1, the equation can then be simplified to

Now that we know that the value of is 18, we can find the value of . To do so, we will again substitute one of the pairs of - and -values from the table into the equation . This time, we will substitute 1 for and 6 for to get

Recall that we just found the value of to be 18, so we can also substitute 18 into the equation for , which gives us

Next, dividing both sides of the equation by 18, we arrive at

Thus, since the value of is 18 and the value of is , an exponential equation in the form for the numbers in the table is .

### Check

To verify our answer, we can substitute the last two -values from the table into our equation to make sure they give us the correct -values. First, substituting 2 for and then simplifying gives us

Next, substituting 3 for and then simplifying, we get

Since both -values give us the correct -value, we can be confident that our equation is correct.

Note that in the process of solving for a, we used one of the properties of
exponents, called the **zero exponent rule**.

### Definition: Zero Exponent Rule

The **zero exponent rule** states that any base raised to the power 0 is equal to 1. That is,
where is the base.

For example, we would use the
**zero exponent rule** to find that .

Now we will write the equation of an exponential function from its graph.

### Example 4: Forming the Equation of an Exponential Function from Its Graph

Observe the given graph, and then answer the following questions

- Find the -intercept in the shown graph.
- As this graph represents an exponential function, every -value is multiplied by when increases by . Find for .
- Find the equation that describes the graph in the form .

### Answer

### Part 1

Let’s begin by finding the graph’s -intercept. We can see that the graph passes through the point on the -axis. In other words, it passes through the -axis at a -value of 10.

Hence, 10 is the -intercept of the shown graph.

### Part 2

Now let’s find the value of in the equation of the graph. In the problem, we are told that the graph represents an exponential function for which every -value is multiplied by when increases by . We are asked to find when .

Keep in mind that means a change of 1 in the value of . Since we have already established that the graph passes through the point , it would be easiest for us to investigate what happens when the -coordinate of a point that the graph passes through increases from 0 to 1. This is a change of 1 in the -value.

We can see that the graph passes through the point , so when the value of changes from 0 to 1, the value of changes from 10 to 20. Remember that we were told that every -value is multiplied by when increases by . Thus, to find the value of when , we must determine what number 10 is multiplied by to get 20. We can do this by dividing 20 by 10:

This tells us that is 2 when .

### Part 3

Now that we know the value of for a specific value of , we can determine the equation that describes the graph in the form . First, we can substitute 2 for and 1 for to get

Next, to find the value of , we can begin by substituting the coordinates of one of the points that the graph passes through into the equation . Here, we will substitute the coordinates of the point into the equation, which gives us

We can then simplify to get

Therefore, we know that the value of is 10. Recall that we also know that is 2 when , so we now have values for , , and that we can substitute into to find the equation of the graph.

Substituting shows us that the equation that describes the graph in the form is , or .

Next, we will work on a problem in which we must determine the value of an expression by evaluating an exponential function.

### Example 5: Evaluating Exponential Functions

Given that , determine the value of .

### Answer

In this problem, we are asked to find the value of an expression that represents the difference of the two fractions and . However, we are only told that and are not given the value of . This means that in order to solve the problem, we must determine the relationship between and . To do this, let’s begin by evaluating the function for some different values of .

1 | 2 | 3 | 4 | |

We can see that for each increase of 1 in the value of , the previous value of is multiplied by 4. That is,

- the value of is ,
- the value of is ,
- the value of is .

In general, we can say that the value of is equal to the value of times 4. This relationship between and will be true for any real number value of .

Since we know that , we also know that . With this knowledge, we can be certain that the value of the fraction is 4 and that the value of the fraction is .

Substituting 4 for and for into the expression given in the problem and simplifying, we get

Thus, given that , we know that the value of must be .

### Note

An alternative solution method involves substituting for both and in the expression given in the problem and then simplifying. Since , we know that . Substituting for and for , we get

Now, simplifying gives us

This is the same value of that we arrived at previously.

### Check

We may be able to identify any possible mistakes by evaluating and at a specific value of . In this case, let’s use . This gives us so the value of our expression becomes

We have already determined that the difference of 4 and is , so we know that is the value of the expression when . This is what we would expect, since the value of the expression should be for any value of .

Note that, in addition to the **quotient rule**, we used the
**negative exponent rule** in our alternative solution. This is another one of the properties of exponents.

### Definition: Negative Exponent Rule

The **negative exponent rule** states that any base raised to a negative power is
equal to 1 over the
base raised to the exponent’s additive inverse. That is,
where is the base and and are the exponents.

For example, we would use the
**negative exponent rule** to find that .

Finally, we will determine the values of an exponential function’s base that result in the function decreasing without being told the form of the function.

### Example 6: Discussing the Monotonicity of Exponential Functions

Consider an exponential function with base . For which values of is the function decreasing?

### Answer

Let’s begin by supposing that we have the exponential function . This function is a monotonic function. That is, it is a function that is always increasing or always decreasing. For this reason, if we can determine that is decreasing over some interval of , then we will know that it is decreasing over all real numbers .

With this in mind, let’s consider the function over the interval . Finding for the integer values of in this interval, we get the following table of values.

1 | 2 | 3 | 4 | |

We can see that for each increase of 1 in the value of , the previous value of is multiplied by . That is,

- the value of is ,
- the value of is ,
- the value of is .

Remember that for any exponential function, the base must be a positive real number not equal to 1. Therefore, to determine the values of for which the function is decreasing, we must ask ourselves, “For what positive real number values of a not equal to 1 is the value of less than the value of ?”

First let’s consider a value of greater than 1. For example, if , then evaluating the function for the integer values of in the interval gives

- ,
- ,
- ,
- .

This shows that if , then the value of is greater than the value of . This will always be true when . In this case, is referred to as the growth factor.

Now let’s consider a value of less than 1. Since must be positive, its value will also be greater than 0. For example, if , then evaluating the function for the integer values of in the interval gives

- ,
- ,
- ,
- .

This shows that if , then the value of is less than the value of . This will always be true when . In this case, is referred to as the decay factor.

Since the value of is less than the value of in the interval when , we know that is decreasing in this interval. Also, because of the function’s monotonicity, we know that if the function is decreasing in this interval, then it is decreasing over all real numbers.

Thus, the values of a for which the function is decreasing are .

### Note

Although we chose as our function, we could have chosen a transformed exponential function and arrived at the same result.

While answering the previous question, we saw that if an exponential function increases or decreases over a certain interval, then it does the same over all real numbers. This is because of its monotonicity.

### Property: Monotonicity of Exponential Functions

An exponential function is a monotonic function, or a function that always increases or always decreases. If the base is greater than 1, it is called the growth factor, and the function will always increase. If the base is greater than 0 and less than 1, it is called the decay factor, and the function will always decrease.

Now let’s finish by recapping some key points.

### Key Points

- An exponential function’s rule has the form , where the constant is the base and the variable is the exponent.
- The base of an exponential function must be a positive real number not equal to 1.
- The
**quotient rule**states that when dividing exponential expressions with the same base, we keep the base and find the difference of the exponents. That is where is the base and and are the exponents. - The
**zero exponent rule**states that any base raised to the power 0 is equal to 1. That is, where is the base. - The
**negative exponent rule**states that any base raised to a negative power is equal to 1 over the base raised to the exponent’s additive inverse. That is, where is the base and and are the exponents. - An exponential function is a monotonic function, or a function that always increases or always decreases.
- If the base of an exponential function is greater than 1, it is called the growth factor, and the function will always increase. If the base is greater than 0 and less than 1, it is called the decay factor, and the function will always decrease.