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מדריכי לימוד > MATH 0123

Exponential Functions

Learning Outcomes

  • Identify the base of an exponential function and restrictions for its value.
  • Find the equation of an exponential function.
  • Use the compound interest formula.
  • Evaluate exponential functions with base e.
  • Given two data points, write an exponential function.
  • Identify initial conditions for an exponential function.
  • Find an exponential function given a graph.
  • Use a graphing calculator to find an exponential function.
  • Find an exponential function that models continuous growth or decay.

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.

Escherichia coli (e Coli) bacteria An electron micrograph of E.Coli bacteria. (credit: “Mattosaurus,” Wikimedia Commons)

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.[footnote]Todar, PhD, Kenneth. Todar's Online Textbook of Bacteriology. http://textbookofbacteriology.net/growth_3.html.[/footnote]

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! If we were to extrapolate the table to twenty-four hours, we would have over 16 million!

Hour 0 1 2 3 4 5 6 7 8 9 10
Bacteria 1 2 4 8 16 32 64 128 256 512 1024
In this module, 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.

Evaluating Exponential Functions

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.[footnote]http://www.worldometers.info/world-population/. Accessed February 24, 2014.[/footnote] 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." 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. 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? This is done to ensure that the outputs will be real numbers. Observe what happens if the base is not positive:
  • Consider a base of –9 and exponent of [latex]\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? This is because a base of 1 results in the constant function. Observe what happens if the base is 1:
  • Consider a base of 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]?

[latex]\begin{array}{llllllll}f\left(x\right)\hfill & ={2}^{x}\hfill & \hfill \\ f\left(3\right)\hfill & ={2}^{3}\text{}\hfill & \text{Substitute }x=3. \hfill \\ \hfill & =8\text{}\hfill & \text{Evaluate the power}\text{.}\hfill \end{array}[/latex]

To evaluate an exponential function, 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]?

[latex]\begin{array}{c}f\left(x\right)\hfill & =30{\left(2\right)}^{x}\hfill & \hfill \\ f\left(3\right)\hfill & =30{\left(2\right)}^{3}\hfill & \text{Substitute }x=3.\hfill \\ \hfill & =30\left(8\right)\text{ }\hfill & \text{Simplify the power first}\text{.}\hfill \\ \hfill & =240\hfill & \text{Multiply}\text{.}\hfill \end{array}[/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]

Example: Evaluating Exponential Functions

Let [latex]f\left(x\right)=5{\left(3\right)}^{x+1}[/latex]. Evaluate [latex]f\left(2\right)[/latex] without using a calculator.

Answer: Follow the order of operations. Be sure to pay attention to the parentheses.

[latex]\begin{array}{c}f\left(x\right)\hfill & =5{\left(3\right)}^{x+1}\hfill & \hfill \\ f\left(2\right)\hfill & =5{\left(3\right)}^{2+1}\hfill & \text{Substitute }x=2.\hfill \\ \hfill & =5{\left(3\right)}^{3}\hfill & \text{Add the exponents}.\hfill \\ \hfill & =5\left(27\right)\hfill & \text{Simplify the power}\text{.}\hfill \\ \hfill & =135\hfill & \text{Multiply}\text{.}\hfill \end{array}[/latex]

Try It

Let [latex]f\left(x\right)=8{\left(1.2\right)}^{x - 5}[/latex]. Evaluate [latex]f\left(3\right)[/latex] using a calculator. Round to four decimal places.

Answer: 5.5556

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.

A General Note: Exponential Growth

A function that models exponential growth grows by a rate proportional to the amount present. For any real number x and any positive real numbers a and b such that [latex]b\ne 1[/latex], an exponential growth function has the form [latex-display]\text{ }f\left(x\right)=a{b}^{x}[/latex-display] where
  • a is the initial or starting value of the function.
  • b is the growth factor or growth multiplier per unit x.
In more general terms, an exponential function consists of a constant base 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.
Year, x Stores, Company A Stores, Company B
0 100 + 50(0) = 100 100(1 + 0.5)0 = 100
1 100 + 50(1) = 150 100(1 + 0.5)1 = 150
2 100 + 50(2) = 200 100(1 + 0.5)2 = 225
3 100 + 50(3) = 250 100(1 + 0.5)3 = 337.5
x A(x) = 100 + 50x B(x) = 100(1 + 0.5)x
The graphs comparing the number of stores for each company over a five-year period are shown below. We can see that, with exponential growth, the number of stores increases much more rapidly than with linear growth.
Graph of Companies A and B’s functions, which values are found in the previous table. 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, 0.5 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.

Example: Evaluating a Real-World Exponential Model

At the beginning of this section, we learned that the population of India was about 1.25 billion in the year 2013 with an annual growth rate of about 1.2%. This situation is represented by the growth function [latex]P\left(t\right)=1.25{\left(1.012\right)}^{t}[/latex] where t is the number of years since 2013. To the nearest thousandth, what will the population of India be in 2031?

Answer: To estimate the population in 2031, we evaluate the models for t = 18, because 2031 is 18 years after 2013. Rounding to the nearest thousandth,

[latex]P\left(18\right)=1.25{\left(1.012\right)}^{18}\approx 1.549[/latex]

There will be about 1.549 billion people in India in the year 2031.

Try It

The population of China was about 1.39 billion in the year 2013 with an annual growth rate of about 0.6%. This situation is represented by the growth function [latex]P\left(t\right)=1.39{\left(1.006\right)}^{t}[/latex] where t is the number of years since 2013. To the nearest thousandth, what will the population of China be in the year 2031? How does this compare to the population prediction we made for India in the previous example?

Answer: About 1.548 billion people; by the year 2031, India’s population will exceed China’s by about 0.001 billion, or 1 million people.

Using 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 compound interest using the compound interest formula which is an exponential function of the variables time t, principal P, APR r, and number of times compounded in a year n:

[latex]A\left(t\right)=P{\left(1+\frac{r}{n}\right)}^{nt}[/latex]

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.
Frequency Value after 1 year
Annually $1100
Semiannually $1102.50
Quarterly $1103.81
Monthly $1104.71
Daily $1105.16

A General Note: The Compound Interest Formula

Compound interest can be calculated using the formula

[latex]A\left(t\right)=P{\left(1+\frac{r}{n}\right)}^{nt}[/latex]

where
  • A(t) is the accumulated value of the account
  • t is measured in years
  • P is the starting amount of the account, often called the principal, or more generally present value
  • r is the annual percentage rate (APR) expressed as a decimal
  • n is the number of times compounded in a year

Example: Calculating Compound Interest

If we invest $3,000 in an investment account paying 3% interest compounded quarterly, how much will the account be worth in 10 years?

Answer: Because we are starting with $3,000, P = 3000. Our interest rate is 3%, so r = 0.03. Because we are compounding quarterly, we are compounding 4 times per year, so n = 4. We want to know the value of the account in 10 years, so we are looking for A(10), the value when t = 10.

[latex]\begin{array}{llllll}A\left(t\right)\hfill & =P\left(1+\frac{r}{n}\right)^{nt}\hfill & \text{Use the compound interest formula}. \\ A\left(10\right)\hfill & =3000\left(1+\frac{0.03}{4}\right)^{4\cdot 10}\hfill & \text{Substitute using given values}. \\ \text{ }\hfill & \approx 4045.05\hfill & \text{Round to two decimal places}.\end{array}[/latex]

The account will be worth about $4,045.05 in 10 years.

Try It

An initial investment of $100,000 at 12% interest is compounded weekly (use 52 weeks in a year). What will the investment be worth in 30 years?

Answer: about $3,644,675.88

Example: Using the Compound Interest Formula to Solve for the Principal

A 529 Plan is a college-savings plan that allows relatives to invest money to pay for a child’s future college tuition; the account grows tax-free. Lily wants to set up a 529 account for her new granddaughter and wants the account to grow to $40,000 over 18 years. She believes the account will earn 6% compounded semi-annually (twice a year). To the nearest dollar, how much will Lily need to invest in the account now?

Answer: The nominal interest rate is 6%, so r = 0.06. Interest is compounded twice a year, so n = 2. We want to find the initial investment P needed so that the value of the account will be worth $40,000 in 18 years. Substitute the given values into the compound interest formula and solve for P.

[latex]\begin{array}{c}A\left(t\right)\hfill & =P{\left(1+\frac{r}{n}\right)}^{nt}\hfill & \text{Use the compound interest formula}.\hfill \\ 40,000\hfill & =P{\left(1+\frac{0.06}{2}\right)}^{2\left(18\right)}\hfill & \text{Substitute using given values }A\text{, }r, n\text{, and }t.\hfill \\ 40,000\hfill & =P{\left(1.03\right)}^{36}\hfill & \text{Simplify}.\hfill \\ \frac{40,000}{{\left(1.03\right)}^{36}}\hfill & =P\hfill & \text{Isolate }P.\hfill \\ P\hfill & \approx 13,801\hfill & \text{Divide and round to the nearest dollar}.\hfill \end{array}[/latex]

Lily will need to invest $13,801 to have $40,000 in 18 years.

Try It

Refer to the previous example. To the nearest dollar, how much would Lily need to invest if the account is compounded quarterly?

Answer: $13,693

Evaluating Exponential Functions with Base e

As we saw earlier, the amount earned on an account increases as the compounding frequency increases. The table below 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.
Frequency [latex]A\left(t\right)={\left(1+\frac{1}{n}\right)}^{n}[/latex] Value
Annually [latex]{\left(1+\frac{1}{1}\right)}^{1}[/latex] $2
Semiannually [latex]{\left(1+\frac{1}{2}\right)}^{2}[/latex] $2.25
Quarterly [latex]{\left(1+\frac{1}{4}\right)}^{4}[/latex] $2.441406
Monthly [latex]{\left(1+\frac{1}{12}\right)}^{12}[/latex] $2.613035
Daily [latex]{\left(1+\frac{1}{365}\right)}^{365}[/latex] $2.714567
Hourly [latex]{\left(1+\frac{1}{\text{8766}}\right)}^{\text{8766}}[/latex] $2.718127
Once per minute [latex]{\left(1+\frac{1}{\text{525960}}\right)}^{\text{525960}}[/latex] $2.718279
Once per second [latex]{\left(1+\frac{1}{31557600}\right)}^{31557600}[/latex] $2.718282
These values appear to be approaching a limit as n increases without bound. In fact, as n gets larger and larger, the expression [latex]{\left(1+\frac{1}{n}\right)}^{n}[/latex] approaches a number used so frequently in mathematics that it has its own name: the letter [latex]e[/latex]. 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.

A General Note: The Number [latex]e[/latex]

The letter e represents the irrational number

[latex]{\left(1+\frac{1}{n}\right)}^{n},\text{as }n\text{ increases without bound}[/latex]

The letter e is used as a base for many real-world exponential models. To work with base e, we use the approximation, [latex]e\approx 2.718282[/latex]. The constant was named by the Swiss mathematician Leonhard Euler (1707–1783) who first investigated and discovered many of its properties.

Example: Using a Calculator to Find Powers of [latex]e[/latex]

Calculate [latex]{e}^{3.14}[/latex]. Round to five decimal places.

Answer: On a calculator, press the button labeled [latex]\left[{e}^{x}\right][/latex]. The window shows [e^(]. Type 3.14 and then close parenthesis, (]). Press [ENTER]. Rounding to 5 decimal places, [latex]{e}^{3.14}\approx 23.10387[/latex]. Caution: Many scientific calculators have an "Exp" button, which is used to enter numbers in scientific notation. It is not used to find powers of e.

Try It

Use a calculator to find [latex]{e}^{-0.5}[/latex]. Round to five decimal places.

Answer: [latex]{e}^{-0.5}\approx 0.60653[/latex]

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, 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{b}^{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{b}^{x}[/latex]. Solve the resulting system of two equations to find [latex]a[/latex] and [latex]b[/latex].
  3. Using the a and b found in the steps above, write the exponential function in the form [latex]f\left(x\right)=a{b}^{x}[/latex].

Example: 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.

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

NOTE: Unless otherwise stated, do not round any intermediate calculations. 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 get larger, the outputs will get increasingly larger resulting in the model not being useful in the long term due to extremely large output values. 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].
Graph of the exponential function, N(t) = 80(1.1447)^t, with labeled points at (0, 80) and (6, 180). 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.

Answer: [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: 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].

Answer: 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{array}{l}6=ab^{-2}\\\frac{6}{b^{-2}}=a\,\,\,\,\,\,\,\,\text{Divide.}\\a=6b^{2}\,\,\,\,\,\,\,\,\text{Use properties of exponents to rewrite the denominator.}\end{array}[/latex]

Substitute a in the second equation and solve for b:

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

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

[latex]a=6b^{2}\approx6\left(0.6389\right)^{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.
Graph of the exponential function, f(x)=2.4492(0.6389)^x, with labeled points at (-2, 6) and (2, 1). 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.

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

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{b}^{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{b}^{x}[/latex]. Solve the resulting system of two equations to find a and b.
  4. Write the exponential function, [latex]f\left(x\right)=a{b}^{x}[/latex].

Example: Writing an Exponential Function Given Its Graph

Find an equation for the exponential function graphed below. Graph of an increasing exponential function with notable points at (0, 3) and (2, 12).

Answer: 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{array}{llllll}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{array}[/latex]

Because we restrict ourselves to positive values of b, we will use b = 2. Substitute a and b into 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 below. Graph of an increasing function with a labeled point at (0, sqrt(2)).

Answer: [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].

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 e 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.

A General Note: The Continuous Growth/Decay Formula

For all real numbers t, and all positive numbers a and r, continuous growth or decay is represented by the formula

[latex]A\left(t\right)=a{e}^{rt}[/latex]

where
  • a is the initial value
  • r is the continuous growth rate per unit of time
  • t is the elapsed time
If r > 0, then the formula represents continuous growth. If r < 0, then the formula represents continuous decay. For business applications, the continuous growth formula is called the continuous compounding formula and takes the form

[latex]A\left(t\right)=P{e}^{rt}[/latex]

where
  • P is the principal or the initial investment
  • r is the growth or interest rate per unit of time
  • t is the period or term of the investment

How To: Given the initial value, rate of growth or decay, and time t, solve a continuous growth or decay function

  1. Use the information in the problem to determine a, the initial value of the function.
  2. Use the information in the problem to determine the growth rate r.
    • If the problem refers to continuous growth, then r > 0.
    • If the problem refers to continuous decay, then r < 0.
  3. Use the information in the problem to determine the time t.
  4. Substitute the given information into the continuous growth formula and solve for A(t).

Example: Calculating Continuous Growth

A person invested $1,000 in an account earning a nominal interest rate of 10% per year compounded continuously. How much was in the account at the end of one year?

Answer: Since the account is growing in value, this is a continuous compounding problem with growth rate r = 0.10. The initial investment was $1,000, so P = 1000. We use the continuous compounding formula to find the value after t = 1 year:

[latex]\begin{array}{c}A\left(t\right)\hfill & =P{e}^{rt}\hfill & \text{Use the continuous compounding formula}.\hfill \\ \hfill & =1000{\left(e\right)}^{0.1} & \text{Substitute known values for }P, r,\text{ and }t.\hfill \\ \hfill & \approx 1105.17\hfill & \text{Use a calculator to approximate}.\hfill \end{array}[/latex]

The account is worth $1,105.17 after one year.

Try It

A person invests $100,000 at a nominal 12% interest per year compounded continuously. What will be the value of the investment in 30 years?

Answer: $3,659,823.44

Example: Calculating Continuous Decay

Radon-222 decays at a continuous rate of 17.3% per day. How much will 100 mg of Radon-222 decay to in 3 days?

Answer: Since the substance is decaying, the rate, 17.3%, is negative. So, r = –0.173. The initial amount of radon-222 was 100 mg, so a = 100. We use the continuous decay formula to find the value after t = 3 days:

[latex]\begin{array}{c}A\left(t\right)\hfill & =a{e}^{rt}\hfill & \text{Use the continuous growth formula}.\hfill \\ \hfill & =100{e}^{-0.173\left(3\right)} & \text{Substitute known values for }a, r,\text{ and }t.\hfill \\ \hfill & \approx 59.5115\hfill & \text{Use a calculator to approximate}.\hfill \end{array}[/latex]

So 59.5115 mg of radon-222 will remain.

Try It

Using the data in the previous example, how much radon-222 will remain after one year?

Answer: 3.77E-26 (This is calculator notation for the number written as [latex]3.77\times {10}^{-26}[/latex] in scientific notation. While the output of an exponential function is never zero, this number is so close to zero that for all practical purposes we can accept zero as the answer.)

Key Equations

definition of the exponential function [latex]f\left(x\right)={b}^{x}\text{, where }b>0, b\ne 1[/latex]
definition of exponential growth [latex]f\left(x\right)=a{b}^{x},\text{ where }a>0,b>0,b\ne 1[/latex]
compound interest formula [latex]\begin{array}{l}A\left(t\right)=P{\left(1+\frac{r}{n}\right)}^{nt} ,\text{ where}\hfill \\ A\left(t\right)\text{ is the account value at time }t\hfill \\ t\text{ is the number of years}\hfill \\ P\text{ is the initial investment, often called the principal}\hfill \\ r\text{ is the annual percentage rate (APR), or nominal rate}\hfill \\ n\text{ is the number of compounding periods in one year}\hfill \end{array}[/latex]
continuous growth formula [latex]A\left(t\right)=a{e}^{rt},\text{ where }[/latex]t is the number of time periods of growth [latex]\\[/latex]a is the starting amount (in the continuous compounding formula a is replaced with P, the principal)[latex]\\[/latex]e is the mathematical constant, [latex] e\approx 2.718282[/latex]

Key Concepts

  • An exponential function is defined as a function with a positive constant other than 1 raised to a variable exponent.
  • A function is evaluated by solving at a specific input value.
  • An exponential model can be found when the growth rate and initial value are known.
  • An exponential model can be found when two data points from the model are known.
  • An exponential model can be found using two data points from the graph of the model.
  • The value of an account at any time t can be calculated using the compound interest formula when the principal, annual interest rate, and compounding periods are known.
  • 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.
  • The number e is a mathematical constant often used as the base of real world exponential growth and decay models. Its decimal approximation is [latex]e\approx 2.718282[/latex].
  • Scientific and graphing calculators have the key [latex]\left[{e}^{x}\right][/latex] or [latex]\left[\mathrm{exp}\left(x\right)\right][/latex] for calculating powers of e.
  • Continuous growth or decay models are exponential models that use e as the base. Continuous growth and decay models can be found when the initial value and growth or decay rate are known.

Glossary

annual percentage rate (APR)
the yearly interest rate earned by an investment account, also called nominal rate
compound interest
interest earned on the total balance, not just the principal
exponential growth
a model that grows by a rate proportional to the amount present
nominal rate
the yearly interest rate earned by an investment account, also called annual percentage rate

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