# Exponential and Logarithmic Functions

## Exponential Functions


### Learning Objectives

1. Find the value of a function (exponential). (IA 3.5.3)
2. Graph exponential functions. (IA 10.2.1)

### Objective 1: Find the value of a function (exponential). (IA 3.5.3)


### Practice Makes Perfect

Find the value of an exponential function.

### Objective 2: Graph exponential functions. (IA 10.2.1)


### Practice Makes Perfect


Graph exponential functions.

The number e, e ≈ 2.718281827, is like the number π 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 or Euler's number after the Swiss mathematician Leonhard Euler.
The exponential function whose base is e,  is called the natural exponential function.

### Practice Makes Perfect

India is the second most populous country in the world with a population of about 
 billion people in 2021. The population is growing at a rate of about 
 each yearhttp://www.worldometers.info/world-population/. Accessed February 24, 2014.. If this rate continues, the population of India will exceed China’s population by the year 
 When populations grow rapidly, we often say that the growth is “exponential,” meaning that something is growing very rapidly. To a mathematician, however, the term exponential growth has a very specific meaning. In this section, we will take a look at exponential functions, which model this kind of rapid growth.

### Identifying Exponential Functions

When exploring linear growth, we observed a constant rate of change—a constant number by which the output increased for each unit increase in input. For example, in the equation 
 the slope tells us the output increases by 3 each time the input increases by 1. The scenario in the India population example is different because we have a percent change per unit time (rather than a constant change) in the number of people.

### Defining an Exponential Function

A study found that the percent of the population who are vegans in the United States doubled from 2009 to 2011. In 2011, 2.5% of the population was vegan, adhering to a diet that does not include any animal products—no meat, poultry, fish, dairy, or eggs. If this rate continues, vegans will make up 10% of the U.S. population in 2015, 40% in 2019, and 80% in 2021.
What exactly does it mean to grow exponentially? What does the word double have in common with percent increase? People toss these words around errantly. Are these words used correctly? The words certainly appear frequently in the media.


For us to gain a clear understanding of exponential growth, let us contrast exponential growth with linear growth. We will construct two functions. The first function is exponential. We will start with an input of 0, and increase each input by 1. We will double the corresponding consecutive outputs. The second function is linear. We will start with an input of 0, and increase each input by 1. We will add 2 to the corresponding consecutive outputs. See .
From  we can infer that for these two functions, exponential growth dwarfs linear growth.
1. Exponential growth refers to the original value from the range increases by the same percentage over equal increments found in the domain.
2. Linear growth refers to the original value from the range increases by the same amount over equal increments found in the domain.
Apparently, the difference between “the same percentage” and “the same amount” is quite significant. For exponential growth, over equal increments, the constant multiplicative rate of change resulted in doubling the output whenever the input increased by one.  For linear growth, the constant additive rate of change over equal increments resulted in adding 2 to the output whenever the input was increased by one.
The general form of the exponential function is 
 where 
 is any nonzero number, 
 is a positive real number not equal to 1.
1. If 
 the function grows at a rate proportional to its size.
2. If 
 the function decays at a rate proportional to its size.
Let’s look at the function 
 from our example. We will create a table () to determine the corresponding outputs over an interval in the domain from 

 to 



Let us examine the graph of 
 by plotting the ordered pairs we observe on the table in , and then make a few observations.
Let’s define the behavior of the graph of the exponential function 
 and highlight some its key characteristics.
1. the domain is 
 
2. the range is 
 
3. as 
 
4. as 
 
5.  
 is always increasing,
6. the graph of 
 will never touch the x-axis because base two raised to any exponent never has the result of zero.
7.  
 is the horizontal asymptote.
8. the y-intercept is 1.

### Evaluating Exponential Functions

Recall that the base of an exponential function must be a positive real number other than 
 Why do we limit the base 
 to positive values? To ensure that the outputs will be real numbers. Observe what happens if the base is not positive:
1. Let 
 and 
 Then 
 which is not a real number.
Why do we limit the base to positive values other than 
 Because base 
 results in the constant function. Observe what happens if the base is 
 
1. Let 
 Then 
 for any value of 
 
To evaluate an exponential function with the form 
 we simply substitute 
 with the given value, and calculate the resulting power. For example:
Let 
 What is 
 
To evaluate an exponential function with a form other than the basic form, it is important to follow the order of operations. For example:
Let 
 What is 
 
Note that if the order of operations were not followed, the result would be incorrect:

### Defining Exponential Growth

Because the output of exponential functions increases very rapidly, the term “exponential growth” is often used in everyday language to describe anything that grows or increases rapidly. However, exponential growth can be defined more precisely in a mathematical sense. If the growth rate is proportional to the amount present, the function models exponential growth.
In more general terms, we have an exponential function, in which a constant base is raised to a variable exponent. To differentiate between linear and exponential functions, let’s consider two companies, A and B. Company A has 100 stores and expands by opening 50 new stores a year, so its growth can be represented by the function 
 Company B has 100 stores and expands by increasing the number of stores by 50% each year, so its growth can be represented by the function 
 
A few years of growth for these companies are illustrated in .
The graphs comparing the number of stores for each company over a five-year period are shown in . We can see that, with exponential growth, the number of stores increases much more rapidly than with linear growth.
Notice that the domain for both functions is 
 and the range for both functions is 
 After year 1, Company B always has more stores than Company A.
Now we will turn our attention to the function representing the number of stores for Company B, 
 In this exponential function, 100 represents the initial number of stores, 0.50 represents the growth rate, and 
 represents the growth factor. Generalizing further, we can write this function as 
 where 100 is the initial value, 
 is called the base, and 
 is called the exponent.

### Finding Equations of Exponential Functions

In the previous examples, we were given an exponential function, which we then evaluated for a given input. Sometimes we are given information about an exponential function without knowing the function explicitly. We must use the information to first write the form of the function, then determine the constants 
 and 
 and evaluate the function.

### Applying the Compound-Interest Formula

Savings instruments in which earnings are continually reinvested, such as mutual funds and retirement accounts, use compound interest. The term compounding refers to interest earned not only on the original value, but on the accumulated value of the account.
The annual percentage rate (APR) of an account, also called the nominal rate, is the yearly interest rate earned by an investment account. The term nominal is used when the compounding occurs a number of times other than once per year. In fact, when interest is compounded more than once a year, the effective interest rate ends up being greater than the nominal rate! This is a powerful tool for investing.
We can calculate the compound interest using the compound interest formula, which is an exponential function of the variables time 
 principal 
 APR 
 and number of compounding periods in a year 
 
For example, observe , which shows the result of investing $1,000 at 10% for one year. Notice how the value of the account increases as the compounding frequency increases.

### Evaluating Functions with Base e

As we saw earlier, the amount earned on an account increases as the compounding frequency increases.  shows that the increase from annual to semi-annual compounding is larger than the increase from monthly to daily compounding. This might lead us to ask whether this pattern will continue.
Examine the value of $1 invested at 100% interest for 1 year, compounded at various frequencies, listed in .
These values appear to be approaching a limit as 
 increases without bound. In fact, as 
 gets larger and larger, the expression 
 approaches a number used so frequently in mathematics that it has its own name: the letter 
 This value is an irrational number, which means that its decimal expansion goes on forever without repeating. Its approximation to six decimal places is shown below.

### Investigating Continuous Growth

So far we have worked with rational bases for exponential functions. For most real-world phenomena, however, e is used as the base for exponential functions. Exponential models that use 
 as the base are called continuous growth or decay models. We see these models in finance, computer science, and most of the sciences, such as physics, toxicology, and fluid dynamics.

### Key Equations


### Key Concepts

1. An exponential function is defined as a function with a positive constant other than 
 raised to a variable exponent. See .
2. A function is evaluated by solving at a specific value. See  and .
3. An exponential model can be found when the growth rate and initial value are known. See .
4. An exponential model can be found when the two data points from the model are known. See .
5. An exponential model can be found using two data points from the graph of the model. See .
6. An exponential model can be found using two data points from the graph and a calculator. See .
7. The value of an account at any time 
 can be calculated using the compound interest formula when the principal, annual interest rate, and compounding periods are known. See .
8. The initial investment of an account can be found using the compound interest formula when the value of the account, annual interest rate, compounding periods, and life span of the account are known. See .
9. The number 
 is a mathematical constant often used as the base of real world exponential growth and decay models. Its decimal approximation is 
 
10. Scientific and graphing calculators have the key 
 or 
 for calculating powers of 
 See .
11. Continuous growth or decay models are exponential models that use 
 as the base. Continuous growth and decay models can be found when the initial value and growth or decay rate are known. See  and .

### Section Exercises


### Verbal


### Algebraic 

For the following exercises, identify whether the statement represents an exponential function. Explain.
For the following exercises, consider this scenario: For each year 
 the population of a forest of trees is represented by the function 
 In a neighboring forest, the population of the same type of tree is represented by the function 
 (Round answers to the nearest whole number.)
For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain.
For the following exercises, find the formula for an exponential function that passes through the two points given.
For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.
For the following exercises, use the compound interest formula, 
 
For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain.

### Numeric

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary.

### Technology 

For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve.

### Extensions 


### Real-World Applications
