Figure 1. Electron micrograph of E.Coli bacteria (credit: “Mattosaurus,” Wikimedia Commons)

Focus in on a square centimeter of your skin. Look closer. Closer still. If you could look closely enough, you would see hundreds of thousands of microscopic organisms. They are bacteria, and they are not only on your skin, but in your mouth, nose, and even your intestines. In fact, the bacterial cells in your body at any given moment outnumber your own cells. But that is no reason to feel bad about yourself. While some bacteria can cause illness, many are healthy and even essential to the body.

Bacteria commonly reproduce through a process called binary fission, during which one bacterial cell splits into two. When conditions are right, bacteria can reproduce very quickly. Unlike humans and other complex organisms, the time required to form a new generation of bacteria is often a matter of minutes or hours, as opposed to days or years.^[1]

For simplicity’s sake, suppose we begin with a culture of one bacterial cell that can divide every hour. The table below shows the number of bacterial cells at the end of each subsequent hour. We see that the single bacterial cell leads to over one thousand bacterial cells in just ten hours! And if we were to extrapolate the table to twenty-four hours, we would have over 16 million!

Exponential Functions

In this chapter, we will explore exponential functions, which can be used for, among other things, modeling growth patterns such as those found in bacteria. We will also investigate logarithmic functions, which are closely related to exponential functions. Both types of functions have numerous real-world applications when it comes to modeling and interpreting data.

India is the second most populous country in the world with a population of about 1.25 billion people in 2013. The population is growing at a rate of about 1.2% each year.^[2] If this rate continues, the population of India will exceed China’s population by the year 2031. 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.

Evaluate Exponential Functions

Recall that the base of an exponential function must be a positive real number other than 1. Why do we limit the base b to positive values? To ensure that the outputs will be real numbers. Observe what happens if the base is not positive:

• Let b = –9 and [latex]x=\frac{1}{2}[/latex]. Then [latex]f\left(x\right)=f\left(\frac{1}{2}\right)={\left(-9\right)}^{\frac{1}{2}}=\sqrt{-9}[/latex], which is not a real number.

Why do we limit the base to positive values other than 1? Because base 1 results in the constant function. Observe what happens if the base is 1:

• Let b = 1. Then [latex]f\left(x\right)={1}^{x}=1[/latex] for any value of x.

To evaluate an exponential function with the form [latex]f\left(x\right)={b}^{x}[/latex], we simply substitute x with the given value, and calculate the resulting power. For example:

Let [latex]f\left(x\right)={2}^{x}[/latex]. What is [latex]f\left(3\right)[/latex]?

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 [latex]f\left(x\right)=30{\left(2\right)}^{x}[/latex]. What is [latex]f\left(3\right)[/latex]?

Note that if the order of operations were not followed, the result would be incorrect:

[latex]f\left(3\right)=30{\left(2\right)}^{3}\ne {60}^{3}=216,000[/latex]

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 [latex]A\left(x\right)=100+50x[/latex]. 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 [latex]B\left(x\right)=100{\left(1+0.5\right)}^{x}[/latex].

A few years of growth for these companies are illustrated below.

The graphs comparing the number of stores for each company over a five-year period are shown in below. We can see that, with exponential growth, the number of stores increases much more rapidly than with linear growth.

Figure 2. The graph shows the numbers of stores Companies A and B opened over a five-year period.

Notice that the domain for both functions is [latex]\left[0,\infty \right)[/latex], and the range for both functions is [latex]\left[100,\infty \right)[/latex]. 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, [latex]B\left(x\right)=100{\left(1+0.5\right)}^{x}[/latex]. In this exponential function, 100 represents the initial number of stores, 0.50 represents the growth rate, and [latex]1+0.5=1.5[/latex] represents the growth factor. Generalizing further, we can write this function as [latex]B\left(x\right)=100{\left(1.5\right)}^{x}[/latex], where 100 is the initial value, 1.5 is called the base, and x is called the exponent.

 Find the Equation of an Exponential Function

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 a and b, and evaluate the function.

How To: Given two data points, write an exponential model.

1. If one of the data points has the form [latex]\left(0,a\right)[/latex], then a is the initial value. Using a, substitute the second point into the equation [latex]f\left(x\right)=a{\left(b\right)}^{x}[/latex], and solve for b.
2. If neither of the data points have the form [latex]\left(0,a\right)[/latex], substitute both points into two equations with the form [latex]f\left(x\right)=a{\left(b\right)}^{x}[/latex]. Solve the resulting system of two equations in two unknowns to find a and b.
3. Using the a and b found in the steps above, write the exponential function in the form [latex]f\left(x\right)=a{\left(b\right)}^{x}[/latex].

Example 3: Writing an Exponential Model When the Initial Value Is Known

In 2006, 80 deer were introduced into a wildlife refuge. By 2012, the population had grown to 180 deer. The population was growing exponentially. Write an algebraic function N(t) representing the population N of deer over time t.

Show Solution

We let our independent variable t be the number of years after 2006. Thus, the information given in the problem can be written as input-output pairs: (0, 80) and (6, 180). Notice that by choosing our input variable to be measured as years after 2006, we have given ourselves the initial value for the function, a = 80. We can now substitute the second point into the equation [latex]N\left(t\right)=80{b}^{t}[/latex] to find b:

[latex]\begin{align}N\left(t\right)& =80{b}^{t} \\ 180& =80{b}^{6} && \text{Substitute using point }\left(6, 180\right). \\ \frac{9}{4} & ={b}^{6} && \text{Divide and write in lowest terms}. \\ b & ={\left(\frac{9}{4}\right)}^{\frac{1}{6}} && \text{Isolate }b\text{ using properties of exponents}. \\ b & \approx 1.1447 && \text{Round to 4 decimal places}. \end{align}[/latex]

NOTE: Unless otherwise stated, do not round any intermediate calculations. Then round the final answer to four places for the remainder of this section.

The exponential model for the population of deer is [latex]N\left(t\right)=80{\left(1.1447\right)}^{t}[/latex]. (Note that this exponential function models short-term growth. As the inputs gets large, the output will get increasingly larger, so much so that the model may not be useful in the long term.)

We can graph our model to observe the population growth of deer in the refuge over time. Notice that the graph below passes through the initial points given in the problem, [latex]\left(0,\text{ 8}0\right)[/latex] and [latex]\left(\text{6},\text{ 18}0\right)[/latex]. We can also see that the domain for the function is [latex]\left[0,\infty \right)[/latex], and the range for the function is [latex]\left[80,\infty \right)[/latex].

Figure 3. Graph showing the population of deer over time, [latex]N\left(t\right)=80{\left(1.1447\right)}^{t}[/latex], t years after 2006

Try It

A wolf population is growing exponentially. In 2011, 129 wolves were counted. By 2013 the population had reached 236 wolves. What two points can be used to derive an exponential equation modeling this situation? Write the equation representing the population N of wolves over time t.

Show Solution

[latex]\left(0,129\right)[/latex] and [latex]\left(2,236\right);N\left(t\right)=129{\left(\text{1}\text{.3526}\right)}^{t}[/latex]

Example 4: Writing an Exponential Model When the Initial Value is Not Known

Find an exponential function that passes through the points [latex]\left(-2,6\right)[/latex] and [latex]\left(2,1\right)[/latex].

Show Solution

Because we don’t have the initial value, we substitute both points into an equation of the form [latex]f\left(x\right)=a{b}^{x}[/latex], and then solve the system for a and b.

• Substituting [latex]\left(-2,6\right)[/latex] gives [latex]6=a{b}^{-2}[/latex]
• Substituting [latex]\left(2,1\right)[/latex] gives [latex]1=a{b}^{2}[/latex]

Use the first equation to solve for a in terms of b:

[latex]\begin{align}&6=ab^{-2}\\&\frac{6}{b^{-2}}=a && \text{Divide.} \\&a=6b^2 && \text{Use properties of exponents to rewrite the denominator.}\end{align}[/latex]

Substitute a in the second equation, and solve for b:

[latex]\begin{align}&1=ab^2 \\ &1=6b^2b^2=6b^4 && \text{Substitute } a=6b^2. \\ &b=\left(\frac{1}{6}\right)^\frac{1}{4} && \text{Use properties of exponents to isolate }b. \\ & b\approx0.6389 && \text{Round to 4 decimal places}. \end{align}[/latex]

Use the value of b in the first equation to solve for the value of a:

[latex]a=6b^2\approx6(0.6389)^2\approx2.4492[/latex]

Thus, the equation is [latex]f\left(x\right)=2.4492{\left(0.6389\right)}^{x}[/latex].

We can graph our model to check our work. Notice that the graph below passes through the initial points given in the problem, [latex]\left(-2,\text{ 6}\right)[/latex] and [latex]\left(2,\text{ 1}\right)[/latex]. The graph is an example of an exponential decay function.

Figure 4. The graph of [latex]f\left(x\right)=2.4492{\left(0.6389\right)}^{x}[/latex] models exponential decay.

Try It

Given the two points [latex]\left(1,3\right)[/latex] and [latex]\left(2,4.5\right)[/latex], find the equation of the exponential function that passes through these two points.

Show Solution

[latex]f\left(x\right)=2{\left(1.5\right)}^{x}[/latex]

Try It

Q & A

Do two points always determine a unique exponential function?

Yes, provided the two points are either both above the x-axis or both below the x-axis and have different x-coordinates. But keep in mind that we also need to know that the graph is, in fact, an exponential function. Not every graph that looks exponential really is exponential. We need to know the graph is based on a model that shows the same percent growth with each unit increase in x, which in many real world cases involves time.

How To: Given the graph of an exponential function, write its equation.

1. First, identify two points on the graph. Choose the y-intercept as one of the two points whenever possible. Try to choose points that are as far apart as possible to reduce round-off error.
2. If one of the data points is the y-intercept [latex]\left(0,a\right)[/latex] , then a is the initial value. Using a, substitute the second point into the equation [latex]f\left(x\right)=a{\left(b\right)}^{x}[/latex], and solve for b.
3. If neither of the data points have the form [latex]\left(0,a\right)[/latex], substitute both points into two equations with the form [latex]f\left(x\right)=a{\left(b\right)}^{x}[/latex]. Solve the resulting system of two equations in two unknowns to find a and b.
4. Write the exponential function, [latex]f\left(x\right)=a{\left(b\right)}^{x}[/latex].

Example 5: Writing an Exponential Function Given Its Graph

Find an equation for the exponential function graphed in Figure 5.

Figure 5

Show Solution

We can choose the y-intercept of the graph, [latex]\left(0,3\right)[/latex], as our first point. This gives us the initial value, [latex]a=3[/latex]. Next, choose a point on the curve some distance away from [latex]\left(0,3\right)[/latex] that has integer coordinates. One such point is [latex]\left(2,12\right)[/latex].

[latex]\begin{align}y&=a{b}^{x} && \text{Write the general form of an exponential equation}. \\ y&=3{b}^{x} && \text{Substitute the initial value 3 for }a. \\ 12&=3{b}^{2} && \text{Substitute in 12 for }y\text{ and 2 for }x. \\ 4&={b}^{2} && \text{Divide by 3}. \\ b&=\pm 2 && \text{Take the square root}.\end{align}[/latex]

Because we restrict ourselves to positive values of b, we will use b = 2. Substitute a and b into the standard form to yield the equation [latex]f\left(x\right)=3{\left(2\right)}^{x}[/latex].

Try It

Find an equation for the exponential function graphed in Figure 6.

Figure 6

Show Solution

[latex]f\left(x\right)=\sqrt{2}{\left(\sqrt{2}\right)}^{x}[/latex]. Answers may vary due to round-off error. The answer should be very close to [latex]1.4142{\left(1.4142\right)}^{x}[/latex].

Try It

How To: Given two points on the curve of an exponential function, use a graphing calculator to find the equation.

1. Press [STAT].
2. Clear any existing entries in columns L1 or L2.
3. In L1, enter the x-coordinates given.
4. In L2, enter the corresponding y-coordinates.
5. Press [STAT] again. Cursor right to CALC, scroll down to ExpReg (Exponential Regression), and press [ENTER].
6. The screen displays the values of a and b in the exponential equation [latex]y=a\cdot {b}^{x}[/latex].

Example 6: Using a Graphing Calculator to Find an Exponential Function

Use a graphing calculator to find the exponential equation that includes the points [latex]\left(2,24.8\right)[/latex] and [latex]\left(5,198.4\right)[/latex].

Show Solution

Follow the guidelines above. First press [STAT], [EDIT], [1: Edit…], and clear the lists L1 and L2. Next, in the L1 column, enter the x-coordinates, 2 and 5. Do the same in the L2 column for the y-coordinates, 24.8 and 198.4.

Now press [STAT], [CALC], [0: ExpReg] and press [ENTER]. The values a = 6.2 and b = 2 will be displayed. The exponential equation is [latex]y=6.2\cdot {2}^{x}[/latex].

Try It

Use a graphing calculator to find the exponential function that includes the points (3, 75.98) and (6, 481.07).

Show Solution

[latex]y\approx 12\cdot {1.85}^{x}[/latex]

 Use Compound Interest Formulas

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 t, principal P, APR r, and number of compounding periods in a year n:

For example, observe the table below, 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.

Evaluate exponential functions with base e