## Fitting Exponential Models to Data

### Learning Objectives

In this section, you will:

• Build an exponential model from data.
• Build a logarithmic model from data.
• Build a logistic model from data.

In previous sections of this chapter, we were either given a function explicitly to graph or evaluate, or we were given a set of points that were guaranteed to lie on the curve. Then we used algebra to find the equation that fit the points exactly. In this section, we use a modeling technique called regression analysis to find a curve that models data collected from real-world observations. With regression analysis, we don’t expect all the points to lie perfectly on the curve. The idea is to find a model that best fits the data. Then we use the model to make predictions about future events.

Do not be confused by the word model. In mathematics, we often use the terms function, equation, and model interchangeably, even though they each have their own formal definition. The term model is typically used to indicate that the equation or function approximates a real-world situation.

We will concentrate on three types of regression models in this section: exponential, logarithmic, and logistic. Having already worked with each of these functions gives us an advantage. Knowing their formal definitions, the behavior of their graphs, and some of their real-world applications gives us the opportunity to deepen our understanding. As each regression model is presented, key features and definitions of its associated function are included for review. Take a moment to rethink each of these functions, reflect on the work we’ve done so far, and then explore the ways regression is used to model real-world phenomena.

### Building an Exponential Model from Data

As we’ve learned, there are a multitude of situations that can be modeled by exponential functions, such as investment growth, radioactive decay, atmospheric pressure changes, and temperatures of a cooling object. What do these phenomena have in common? For one thing, all the models either increase or decrease as time moves forward. But that’s not the whole story. It’s the way data increase or decrease that helps us determine whether it is best modeled by an exponential equation. Knowing the behavior of exponential functions in general allows us to recognize when to use exponential regression, so let’s review exponential growth and decay.

Recall that exponential functions have the form$\,y=a{b}^{x}\,$or$\,y={A}_{0}{e}^{kx}.\,$When performing regression analysis, we use the form most commonly used on graphing utilities,$\,y=a{b}^{x}.\,$Take a moment to reflect on the characteristics we’ve already learned about the exponential function$\,y=a{b}^{x}\,$(assume$\,a>0$):

• $b\,$must be greater than zero and not equal to one.
• The initial value of the model is$\,y=a.$
• If$\,b>1,$ the function models exponential growth. As$\,x\,$increases, the outputs of the model increase slowly at first, but then increase more and more rapidly, without bound.
• If$\,0<b<1,$ the function models exponential decay. As$\,x\,$increases, the outputs for the model decrease rapidly at first and then level off to become asymptotic to the x-axis. In other words, the outputs never become equal to or less than zero.

As part of the results, your calculator will display a number known as the correlation coefficient, labeled by the variable$\,r,$ or$\,{r}^{2}.\,$(You may have to change the calculator’s settings for these to be shown.) The values are an indication of the “goodness of fit” of the regression equation to the data. We more commonly use the value of$\,{r}^{2}\,$instead of$\,r,$ but the closer either value is to 1, the better the regression equation approximates the data.

### Exponential Regression

Exponential regression is used to model situations in which growth begins slowly and then accelerates rapidly without bound, or where decay begins rapidly and then slows down to get closer and closer to zero. We use the command “ExpReg” on a graphing utility to fit an exponential function to a set of data points. This returns an equation of the form,$y=a{b}^{x}$

Note that:

• $b\,$must be non-negative.
• when$\,b>1,$ we have an exponential growth model.
• when$\,0<b<1,$ we have an exponential decay model.

### How To

Given a set of data, perform exponential regression using a graphing utility.

1. Use the STAT then EDIT menu to enter given data.
1. Clear any existing data from the lists.
2. List the input values in the L1 column.
3. List the output values in the L2 column.
2. Graph and observe a scatter plot of the data using the STATPLOT feature.
1. Use ZOOM [9] to adjust axes to fit the data.
2. Verify the data follow an exponential pattern.
3. Find the equation that models the data.
1. Select “ExpReg” from the STAT then CALC menu.
2. Use the values returned for a and b to record the model,$\,y=a{b}^{x}.$
4. Graph the model in the same window as the scatterplot to verify it is a good fit for the data.

### Using Exponential Regression to Fit a Model to Data

In 2007, a university study was published investigating the crash risk of alcohol impaired driving. Data from 2,871 crashes were used to measure the association of a person’s blood alcohol level (BAC) with the risk of being in an accident. (Figure) shows results from the study[1] . The relative risk is a measure of how many times more likely a person is to crash. So, for example, a person with a BAC of 0.09 is 3.54 times as likely to crash as a person who has not been drinking alcohol.

 BAC 0 0.01 0.03 0.05 0.07 0.09 Relative Risk of Crashing 1 1.03 1.06 1.38 2.09 3.54 BAC 0.11 0.13 0.15 0.17 0.19 0.21 Relative Risk of Crashing 6.41 12.6 22.1 39.05 65.32 99.78
1. Let$\,x\,$represent the BAC level, and let$\,y\,$represent the corresponding relative risk. Use exponential regression to fit a model to these data.
2. After 6 drinks, a person weighing 160 pounds will have a BAC of about$\,0.16.\,$How many times more likely is a person with this weight to crash if they drive after having a 6-pack of beer? Round to the nearest hundredth.

### Try It

(Figure) shows a recent graduate’s credit card balance each month after graduation.

Hsu-Mei wants to save $5,000 for a down payment on a car. To the nearest dollar, how much will she need to invest in an account now with$\,7.5%\,$APR, compounded daily, in order to reach her goal in$\,3\,$years? Does the equation$\,y=2.294{e}^{-0.654t}\,$represent continuous growth, continuous decay, or neither? Explain. Suppose an investment account is opened with an initial deposit of$\,\text{\10,500}\,$earning$\,6.25%\,$interest, compounded continuously. How much will the account be worth after$\,25\,$years? #### Graphs of Exponential Functions Graph the function$\,f\left(x\right)=3.5{\left(2\right)}^{x}.\,$State the domain and range and give the y-intercept. Graph the function$\,f\left(x\right)=4{\left(\frac{1}{8}\right)}^{x}\,$and its reflection about the y-axis on the same axes, and give the y-intercept. The graph of$\,f\left(x\right)={6.5}^{x}\,$is reflected about the y-axis and stretched vertically by a factor of$\,7.\,$What is the equation of the new function,$\,g\left(x\right)?\,$State its y-intercept, domain, and range. The graph below shows transformations of the graph of$\,f\left(x\right)={2}^{x}.\,$What is the equation for the transformation? #### Logarithmic Functions Rewrite$\,{\mathrm{log}}_{17}\left(4913\right)=x\,$as an equivalent exponential equation. Rewrite$\,\mathrm{ln}\left(s\right)=t\,$as an equivalent exponential equation. Rewrite$\,{a}^{-\,\frac{2}{5}}=b\,$as an equivalent logarithmic equation. Rewrite $\,{e}^{-3.5}=h\,$ as an equivalent logarithmic equation. Solve for x if$\,\,\,{\mathrm{log}}_{64}\left(x\right)=\frac{1}{3}\,$by converting to exponential form. Evaluate$\,{\mathrm{log}}_{5}\left(\frac{1}{125}\right)\,$without using a calculator. Evaluate$\,\mathrm{log}\left(\text{0}\text{.000001}\right)\,$without using a calculator. Evaluate$\,\mathrm{log}\left(4.005\right)\,$using a calculator. Round to the nearest thousandth. Evaluate$\,\mathrm{ln}\left({e}^{-0.8648}\right)\,$without using a calculator. Evaluate$\,\mathrm{ln}\left(\sqrt[3]{18}\right)\,$using a calculator. Round to the nearest thousandth. #### Graphs of Logarithmic Functions Graph the function$\,g\left(x\right)=\mathrm{log}\left(7x+21\right)-4.$ Graph the function$\,h\left(x\right)=2\mathrm{ln}\left(9-3x\right)+1.$ State the domain, vertical asymptote, and end behavior of the function$\,g\left(x\right)=\mathrm{ln}\left(4x+20\right)-17.$ #### Logarithmic Properties Rewrite$\,\mathrm{ln}\left(7r\cdot 11st\right)\,$in expanded form. Rewrite$\,{\mathrm{log}}_{8}\left(x\right)+{\mathrm{log}}_{8}\left(5\right)+{\mathrm{log}}_{8}\left(y\right)+{\mathrm{log}}_{8}\left(13\right)\,$in compact form. Rewrite$\,{\mathrm{log}}_{m}\left(\frac{67}{83}\right)\,$in expanded form. Rewrite$\,\mathrm{ln}\left(z\right)-\mathrm{ln}\left(x\right)-\mathrm{ln}\left(y\right)\,$in compact form. Rewrite$\,\mathrm{ln}\left(\frac{1}{{x}^{5}}\right)\,$as a product. Rewrite$\,-{\mathrm{log}}_{y}\left(\frac{1}{12}\right)\,$as a single logarithm. Use properties of logarithms to expand$\,\mathrm{log}\left(\frac{{r}^{2}{s}^{11}}{{t}^{14}}\right).$ Use properties of logarithms to expand$\,\mathrm{ln}\left(2b\sqrt{\frac{b+1}{b-1}}\right).$ Condense the expression$\,5\mathrm{ln}\left(b\right)+\mathrm{ln}\left(c\right)+\frac{\mathrm{ln}\left(4-a\right)}{2}\,$to a single logarithm. Condense the expression$\,3{\mathrm{log}}_{7}v+6{\mathrm{log}}_{7}w-\frac{{\mathrm{log}}_{7}u}{3}\,$to a single logarithm. Rewrite$\,{\mathrm{log}}_{3}\left(12.75\right)\,$to base$\,e.$ Rewrite$\,{5}^{12x-17}=125\,$as a logarithm. Then apply the change of base formula to solve for$\,x\,$using the common log. Round to the nearest thousandth. #### Exponential and Logarithmic Equations Solve$\,{216}^{3x}\cdot {216}^{x}={36}^{3x+2}\,$by rewriting each side with a common base. Solve$\,\frac{125}{{\left(\frac{1}{625}\right)}^{-x-3}}={5}^{3}\,$by rewriting each side with a common base. Use logarithms to find the exact solution for$\,7\cdot {17}^{-9x}-7=49.\,$If there is no solution, write no solution. Use logarithms to find the exact solution for$\,3{e}^{6n-2}+1=-60.\,$If there is no solution, write no solution. Find the exact solution for$\,5{e}^{3x}-4=6\,$. If there is no solution, write no solution. Find the exact solution for$\,2{e}^{5x-2}-9=-56.\,$If there is no solution, write no solution. Find the exact solution for$\,{5}^{2x-3}={7}^{x+1}.\,$If there is no solution, write no solution. Find the exact solution for$\,{e}^{2x}-{e}^{x}-110=0.\,$If there is no solution, write no solution. Use the definition of a logarithm to solve.$\,-5{\mathrm{log}}_{7}\left(10n\right)=5.$ 47. Use the definition of a logarithm to find the exact solution for$\,9+6\mathrm{ln}\left(a+3\right)=33.$ Use the one-to-one property of logarithms to find an exact solution for$\,{\mathrm{log}}_{8}\left(7\right)+{\mathrm{log}}_{8}\left(-4x\right)={\mathrm{log}}_{8}\left(5\right).\,$If there is no solution, write no solution. Use the one-to-one property of logarithms to find an exact solution for$\,\mathrm{ln}\left(5\right)+\mathrm{ln}\left(5{x}^{2}-5\right)=\mathrm{ln}\left(56\right).\,$If there is no solution, write no solution. The formula for measuring sound intensity in decibels$\,D\,$is defined by the equation$\,D=10\mathrm{log}\left(\frac{I}{{I}_{0}}\right),$ where$\,I\,$is the intensity of the sound in watts per square meter and$\,{I}_{0}={10}^{-12}\,$is the lowest level of sound that the average person can hear. How many decibels are emitted from a large orchestra with a sound intensity of$\,6.3\cdot {10}^{-3}\,$watts per square meter? The population of a city is modeled by the equation$\,P\left(t\right)=256,114{e}^{0.25t}\,$where$\,t\,$is measured in years. If the city continues to grow at this rate, how many years will it take for the population to reach one million? Find the inverse function$\,{f}^{-1}\,$for the exponential function$\,f\left(x\right)=2\cdot {e}^{x+1}-5.$ Find the inverse function$\,{f}^{-1}\,$for the logarithmic function$\,f\left(x\right)=0.25\cdot {\mathrm{log}}_{2}\left({x}^{3}+1\right).$ #### Exponential and Logarithmic Models For the following exercises, use this scenario: A doctor prescribes$\,300\,$milligrams of a therapeutic drug that decays by about$\,17%\,$each hour. To the nearest minute, what is the half-life of the drug? Write an exponential model representing the amount of the drug remaining in the patient’s system after$\,t\,$hours. Then use the formula to find the amount of the drug that would remain in the patient’s system after$\,24\,$hours. Round to the nearest hundredth of a gram. For the following exercises, use this scenario: A soup with an internal temperature of$\,\text{350°}\,$Fahrenheit was taken off the stove to cool in a$\,\text{71°F}\,$room. After fifteen minutes, the internal temperature of the soup was$\,\text{175°F}\text{.}$ Use Newton’s Law of Cooling to write a formula that models this situation. How many minutes will it take the soup to cool to$\,\text{85°F?}$ For the following exercises, use this scenario: The equation$\,N\left(t\right)=\frac{1200}{1+199{e}^{-0.625t}}\,$models the number of people in a school who have heard a rumor after$\,t\,$days. How many people started the rumor? To the nearest tenth, how many days will it be before the rumor spreads to half the carrying capacity? What is the carrying capacity? For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic.  x f(x) 1 3.05 2 4.42 3 6.4 4 9.28 5 13.46 6 19.52 7 28.3 8 41.04 9 59.5 10 86.28  x f(x) 0.5 18.05 1 17 3 15.33 5 14.55 7 14.04 10 13.5 12 13.22 13 13.1 15 12.88 17 12.69 20 12.45 Find a formula for an exponential equation that goes through the points$\,\left(-2,100\right)\,$and$\,\left(0,4\right).\,$Then express the formula as an equivalent equation with base e. #### Fitting Exponential Models to Data What is the carrying capacity for a population modeled by the logistic equation$\,P\left(t\right)=\frac{250,000}{1\,\,+\,\,499{e}^{-0.45t}}?\,$What is the initial population for the model? The population of a culture of bacteria is modeled by the logistic equation$\,P\left(t\right)=\frac{14,250}{1\,\,+\,\,29{e}^{-0.62t}},$ where$\,t\,$is in days. To the nearest tenth, how many days will it take the culture to reach$\,75%\,$of its carrying capacity? For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.  x f(x) 1 409.4 2 260.7 3 170.4 4 110.6 5 74 6 44.7 7 32.4 8 19.5 9 12.7 10 8.1  x f(x) 0.15 36.21 0.25 28.88 0.5 24.39 0.75 18.28 1 16.5 1.5 12.99 2 9.91 2.25 8.57 2.75 7.23 3 5.99 3.5 4.81  x f(x) 0 9 2 22.6 4 44.2 5 62.1 7 96.9 8 113.4 10 133.4 11 137.6 15 148.4 17 149.3 ### Practice Test The population of a pod of bottlenose dolphins is modeled by the function$\,A\left(t\right)=8{\left(1.17\right)}^{t},$ where$\,t\,$is given in years. To the nearest whole number, what will the pod population be after$\,3\,$years? Find an exponential equation that passes through the points$\,\text{(0, 4)}\,$and$\,\text{(2, 9)}\text{.}$ Drew wants to save$2,500 to go to the next World Cup. To the nearest dollar, how much will he need to invest in an account now with$\,6.25%\,$APR, compounding daily, in order to reach his goal in$\,4\,$years?

An investment account was opened with an initial deposit of \$9,600 and earns$\,7.4%\,$interest, compounded continuously. How much will the account be worth after$\,15\,$years?

Graph the function$\,f\left(x\right)=5{\left(0.5\right)}^{-x}\,$and its reflection across the y-axis on the same axes, and give the y-intercept.

The graph shows transformations of the graph of$\,f\left(x\right)={\left(\frac{1}{2}\right)}^{x}.\,$What is the equation for the transformation?

Rewrite$\,{\mathrm{log}}_{8.5}\left(614.125\right)=a\,$as an equivalent exponential equation.

Rewrite$\,{e}^{\frac{1}{2}}=m\,$as an equivalent logarithmic equation.

Solve for$\,x\,$by converting the logarithmic equation$\,lo{g}_{\frac{1}{7}}\left(x\right)=2\,$to exponential form.

Evaluate$\,\mathrm{log}\left(\text{10,000,000}\right)\,$without using a calculator.

Evaluate$\,\mathrm{ln}\left(0.716\right)\,$using a calculator. Round to the nearest thousandth.

Graph the function$\,g\left(x\right)=\mathrm{log}\left(12-6x\right)+3.$

State the domain, vertical asymptote, and end behavior of the function$\,f\left(x\right)={\mathrm{log}}_{5}\left(39-13x\right)+7.$

Rewrite$\,\mathrm{log}\left(17a\cdot 2b\right)\,$as a sum.

Rewrite$\,{\mathrm{log}}_{t}\left(96\right)-{\mathrm{log}}_{t}\left(8\right)\,$in compact form.

Rewrite$\,{\mathrm{log}}_{8}\left({a}^{\frac{1}{b}}\right)\,$as a product.

Use properties of logarithm to expand$\,\mathrm{ln}\left({y}^{3}{z}^{2}\cdot \sqrt[3]{x-4}\right).$

Condense the expression$\,4\mathrm{ln}\left(c\right)+\mathrm{ln}\left(d\right)+\frac{\mathrm{ln}\left(a\right)}{3}+\frac{\mathrm{ln}\left(b+3\right)}{3}\,$to a single logarithm.

Rewrite$\,{16}^{3x-5}=1000\,$as a logarithm. Then apply the change of base formula to solve for$\,x\,$using the natural log. Round to the nearest thousandth.

Solve$\,{\left(\frac{1}{81}\right)}^{x}\cdot \frac{1}{243}={\left(\frac{1}{9}\right)}^{-3x-1}\,$by rewriting each side with a common base.

Use logarithms to find the exact solution for$\,-9{e}^{10a-8}-5=-41$. If there is no solution, write no solution.

Find the exact solution for$\,10{e}^{4x+2}+5=56.\,$If there is no solution, write no solution.

Find the exact solution for$\,-5{e}^{-4x-1}-4=64.\,$If there is no solution, write no solution.

Find the exact solution for$\,{2}^{x-3}={6}^{2x-1}.\,$If there is no solution, write no solution.

Find the exact solution for$\,{e}^{2x}-{e}^{x}-72=0.\,$If there is no solution, write no solution.

Use the definition of a logarithm to find the exact solution for$\,4\mathrm{log}\left(2n\right)-7=-11$

Use the one-to-one property of logarithms to find an exact solution for$\,\mathrm{log}\left(4{x}^{2}-10\right)+\mathrm{log}\left(3\right)=\mathrm{log}\left(51\right)\,$If there is no solution, write no solution.

The formula for measuring sound intensity in decibels$\,D\,$is defined by the equation$\,D=10\mathrm{log}\left(\frac{I}{{I}_{0}}\right),$where$\,I\,$is the intensity of the sound in watts per square meter and$\,{I}_{0}={10}^{-12}\,$is the lowest level of sound that the average person can hear. How many decibels are emitted from a rock concert with a sound intensity of$\,4.7\cdot {10}^{-1}\,$watts per square meter?

A radiation safety officer is working with$\,112\,$grams of a radioactive substance. After$\,17\,$days, the sample has decayed to$\,80\,$grams. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest day, what is the half-life of this substance?

Write the formula found in the previous exercise as an equivalent equation with base$\,e.\,$Express the exponent to five significant digits.

A bottle of soda with a temperature of$\,\text{71°}\,$Fahrenheit was taken off a shelf and placed in a refrigerator with an internal temperature of$\,\text{35° F}\text{.}\,$After ten minutes, the internal temperature of the soda was$\,\text{63° F}\text{.}\,$Use Newton’s Law of Cooling to write a formula that models this situation. To the nearest degree, what will the temperature of the soda be after one hour?

The population of a wildlife habitat is modeled by the equation$\,P\left(t\right)=\frac{360}{1+6.2{e}^{-0.35t}},$ where$\,t\,$is given in years. How many animals were originally transported to the habitat? How many years will it take before the habitat reaches half its capacity?

Enter the data from (Figure) into a graphing calculator and graph the resulting scatter plot. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic.

 x f(x) 1 3 2 8.55 3 11.79 4 14.09 5 15.88 6 17.33 7 18.57 8 19.64 9 20.58 10 21.42

The population of a lake of fish is modeled by the logistic equation$\,P\left(t\right)=\frac{16,120}{1+25{e}^{-0.75t}},$ where$\,t\,$is time in years. To the nearest hundredth, how many years will it take the lake to reach$\,80%\,$of its carrying capacity?

For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.

 x f(x) 1 20 2 21.6 3 29.2 4 36.4 5 46.6 6 55.7 7 72.6 8 87.1 9 107.2 10 138.1

 x f(x) 3 13.98 4 17.84 5 20.01 6 22.7 7 24.1 8 26.15 9 27.37 10 28.38 11 29.97 12 31.07 13 31.43
 x f(x) 0 2.2 0.5 2.9 1 3.9 1.5 4.8 2 6.4 3 9.3 4 12.3 5 15 6 16.2 7 17.3 8 17.9

1. •Source: Indiana University Center for Studies of Law in Action, 2007
2. •Source: Center for Disease Control and Prevention, 2013
3. •Source: The World Bank, 2013