# Exponential and Logarithmic Functions

## Evaluate and Graph Exponential Functions


### Graph Exponential Functions

The functions we have studied so far do not give us a model for many naturally occurring phenomena. From the growth of populations and the spread of viruses to radioactive decay and compounding interest, the models are very different from what we have studied so far. These models involve exponential functions.
An exponential function is a function of the form  where  and 
Notice that in this function, the variable is the exponent. In our functions so far, the variables were the base.
Our definition says  If we let  then  becomes  Since  for all real numbers,  This is the constant function.
Our definition also says  If we let a base be negative, say  then  is not a real number when 
In fact,  would not be a real number any time  is a fraction with an even denominator. So our definition requires 
By graphing a few exponential functions, we will be able to see their unique properties.
If we look at the graphs from the previous Example and Try Its, we can identify some of the properties of exponential functions.
The graphs of  and  as well as the graphs of  and  all have the same basic shape. This is the shape we expect from an exponential function where 
We notice, that for each function, the graph contains the point  This make sense because  for any a.
The graph of each function,  also contains the point  The graph of  contained  and the graph of  contained  This makes sense as 
Notice too, the graph of each function  also contains the point  The graph of  contained  and the graph of  contained  This makes sense as 
What is the domain for each function? From the graphs we can see that the domain is the set of all real numbers. There is no restriction on the domain. We write the domain in interval notation as 
Look at each graph. What is the range of the function? The graph never hits the -axis. The range is all positive numbers. We write the range in interval notation as 
Whenever a graph of a function approaches a line but never touches it, we call that line an asymptote. For the exponential functions we are looking at, the graph approaches the -axis very closely but will never cross it, we call the line  the x-axis, a horizontal asymptote.
Our definition of an exponential function  says  but the examples and discussion so far has been about functions where  What happens when ? The next example will explore this possibility.
Now let’s look at the graphs from the previous Example and Try Its so we can now identify some of the properties of exponential functions where 
The graphs of  and  as well as the graphs of  and  all have the same basic shape. While this is the shape we expect from an exponential function where  the graphs go down from left to right while the previous graphs, when  went from up from left to right.
We notice that for each function, the graph still contains the point (0, 1). This make sense because  for any a.
As before, the graph of each function,  also contains the point  The graph of  contained  and the graph of  contained  This makes sense as 
Notice too that the graph of each function,  also contains the point  The graph of  contained  and the graph of  contained  This makes sense as 
What is the domain and range for each function? From the graphs we can see that the domain is the set of all real numbers and we write the domain in interval notation as  Again, the graph never hits the -axis. The range is all positive numbers. We write the range in interval notation as 
We will summarize these properties in the chart below. Which also include when 
It is important for us to notice that both of these graphs are one-to-one, as they both pass the horizontal line test. This means the exponential function will have an inverse. We will look at this later.
When we graphed quadratic functions, we were able to graph using translation rather than just plotting points. Will that work in graphing exponential functions?
Looking at the graphs of the functions  and  in the last example, we see that adding one in the exponent caused a horizontal shift of one unit to the left. Recognizing this pattern allows us to graph other functions with the same pattern by translation.
Let’s now consider another situation that might be graphed more easily by translation, once we recognize the pattern.
Looking at the graphs of the functions  and  in the last example, we see that subtracting 2 caused a vertical shift of down two units. Notice that the horizontal asymptote also shifted down 2 units. Recognizing this pattern allows us to graph other functions with the same pattern by translation.
All of our exponential functions have had either an integer or a rational number as the base. We will now look at an exponential function with an irrational number as the base.
Before we can look at this exponential function, we need to define the irrational number, e. This number is used as a base in many applications in the sciences and business that are modeled by exponential functions. The number is defined as the value of  as n gets larger and larger. We say, as n approaches infinity, or increases without bound. The table shows the value of  for several values of 
The number e is like the number  in that we use a symbol to represent it because its decimal representation never stops or repeats. The irrational number e is called the natural base.
The exponential function whose base is   is called the natural exponential function.
Let’s graph the function  on the same coordinate system as  and 
Notice that the graph of  is “between” the graphs of  and  Does this make sense as ?

### Solve Exponential Equations

Equations that include an exponential expression  are called exponential equations. To solve them we use a property that says as long as  and  if  then it is true that  In other words, in an exponential equation, if the bases are equal then the exponents are equal.
To use this property, we must be certain that both sides of the equation are written with the same base.
The steps are summarized below.
In the next example, we will use our properties on exponents.

### Use Exponential Models in Applications

Exponential functions model many situations. If you own a bank account, you have experienced the use of an exponential function. There are two formulas that are used to determine the balance in the account when interest is earned. If a principal, P, is invested at an interest rate, r, for t years, the new balance, A, will depend on how often the interest is compounded. If the interest is compounded n times a year we use the formula  If the interest is compounded continuously, we use the formula  These are the formulas for compound interest.
As you work with the Interest formulas, it is often helpful to identify the values of the variables first and then substitute them into the formula.
Other topics that are modeled by exponential functions involve growth and decay. Both also use the formula  we used for the growth of money. For growth and decay, generally we use as the original amount instead of calling it  the principal. We see that exponential growth has a positive rate of growth and exponential decay has a negative rate of growth.
Exponential growth is typically seen in the growth of populations of humans or animals or bacteria. Our next example looks at the growth of a virus.

### Key Concepts

1. Properties of the Graph of 





2. One-to-One Property of Exponential Functions:
For  and 

3. How to Solve an Exponential Equation


4. Compound Interest: For a principal,  invested at an interest rate,  for  years, the new balance,  is

5. Exponential Growth and Decay: For an original amount,  that grows or decays at a rate,  for a certain time  the final amount, is 

### Practice Makes Perfect

Graph Exponential Functions
In the following exercises, graph each exponential function.
In the following exercises, graph each function in the same coordinate system.
In the following exercises, graph each exponential function.
Solve Exponential Equations
In the following exercises, solve each equation.
In the following exercises, match the graphs to one of the following functions: ⓐ  ⓑ  ⓒ  ⓓ  ⓔ  ⓕ 
Use exponential models in applications
In the following exercises, use an exponential model to solve.

### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?