MTH120 College Algebra Chapter 6.8

 6.8 Fitting Exponential Models to Data

Fitting exponential models to data is a common task in data analysis and modeling. Exponential models are often used to describe processes with exponential growth or decay. Here's a general approach to fitting exponential models to data:

1. Collect Data: Gather the data that you want to analyze. Ensure that the data represents a process that exhibits exponential growth or decay.

2. Understand the Model: Know the form of the exponential model you intend to fit to the data. Exponential growth and decay models have the form $�=�\cdot {�}^{�}$, where $�$ and $�$ are parameters.

3. Linearize the Model: Exponential models can often be linearized by taking the natural logarithm (ln) of both sides. The linearized form becomes $\ln (�)=\ln (�)+�\cdot \ln (�)$. This allows you to use linear regression techniques to fit the model.

4. Perform Linear Regression: Use a statistical software package or spreadsheet program to perform linear regression on the linearized model. Fit a straight line to the data points in the form ${�}^{\mathrm{\prime }}=�+��$, where $�$ is the intercept, and $�$ is the slope.

5. Interpret the Parameters: After performing linear regression, you will obtain values for the parameters $�$ and $�$. These correspond to $\ln (�)$ and $\ln (�)$ in the original exponential model. To find the values of $�$ and $�$, you can exponentiate the results: $�={�}^{�}$ and $�={�}^{�}$.

6. Evaluate the Fit: Assess the quality of the fit by analyzing the goodness of fit statistics, such as R-squared, p-values, and residuals. A higher R-squared value indicates a better fit.

7. Make Predictions: Once you have the parameter values for the exponential model, you can use the model to make predictions for values of $�$ that were not in the original dataset.

8. Plot the Fit: Plot the fitted exponential model along with the original data points to visualize how well the model represents the data.

It's important to note that not all data follows an exact exponential model. It's essential to assess the appropriateness of the model for your specific data and understand the underlying assumptions. Additionally, if the data exhibits significant deviations from the exponential model, other models or more complex functions may be needed.

The steps above outline the process for fitting a simple exponential model to data. In some cases, you may encounter variations of exponential models that require different techniques, or you may use specialized software for curve fitting.

Building an exponential model from data involves finding the best-fitting exponential equation that describes the relationship between the variables in your dataset. Here's a step-by-step guide with examples:

Step 1: Collect and Examine the Data

• Collect the data that represents the phenomenon you want to model.
• Examine the data to determine if it exhibits exponential growth or decay. For exponential growth, the data should be increasing at an increasing rate, and for exponential decay, it should be decreasing at a decreasing rate.

Step 2: Choose the Appropriate Exponential Model

• Decide whether you're dealing with exponential growth or decay and choose the appropriate form of the exponential equation: $�=�\cdot {�}^{�}$ for growth or $�=�\cdot {�}^{-�}$ for decay. Here, $�$ and $�$ are parameters you need to determine.

Step 3: Linearize the Model

• In most cases, you'll want to linearize the model by taking the natural logarithm of both sides. For exponential growth, the linearized equation is $\ln (�)=\ln (�)+�\cdot \ln (�)$, and for exponential decay, it's $\ln (�)=\ln (�)-�\cdot \ln (�)$. This linearized form simplifies the fitting process.

Step 4: Perform Linear Regression

• Use a statistical software package or spreadsheet program to perform linear regression on the linearized model. Fit a straight line to the data points, and you'll obtain the intercept and slope of the line.

Step 5: Interpret the Parameters

• The intercept and slope from the linear regression represent $\ln (�)$ and $\ln (�)$, respectively. To find the values of $�$ and $�$, you can exponentiate the results: $�={�}^{\text{intercept}}$ and $�={�}^{\text{slope}}$.

Step 6: Evaluate the Fit

• Assess the quality of the fit by analyzing the goodness of fit statistics, such as R-squared, p-values, and residuals. A higher R-squared value indicates a better fit.

Step 7: Make Predictions

• Use the parameters $�$ and $�$ to make predictions for values of $�$ that were not in the original dataset. Plug these values into the exponential equation to estimate corresponding $�$ values.

Step 8: Plot the Fit

• Create a plot that displays the fitted exponential model along with the original data points. This visualization helps you see how well the model represents the data.

Now, let's work through an example of building an exponential growth model from data:

Example: Population Growth of Bacteria Suppose you have the following data representing the population of bacteria over time:

Time (hours)	Population
0	100
1	200
2	400
3	800
4	1600

You want to build an exponential growth model from this data.

Step 1: Collect and Examine the Data The data represents exponential growth because the population is doubling at each time step.

Step 2: Choose the Appropriate Exponential Model We're dealing with exponential growth, so the appropriate model is $�=�\cdot {�}^{�}$.

Step 3: Linearize the Model Taking the natural logarithm of the model, we get $\ln (�)=\ln (�)+�\cdot \ln (�)$.

Step 4: Perform Linear Regression Perform linear regression on the linearized model using software. The result is $\ln (�)\approx 4.6052$ and $\ln (�)\approx 0.6931$.

Step 5: Interpret the Parameters Exponentiating the results: $�\approx {�}^{4.6052}\approx 100.0$ and $�\approx {�}^{0.6931}\approx 2.0$.

Step 6: Evaluate the Fit Assess the quality of the fit using R-squared and other statistics.

Step 7: Make Predictions You can now use the model $�=100\cdot 2^{�}$ to predict the population at any time.

Step 8: Plot the Fit Create a plot that shows the fitted exponential model and the original data points to visualize the fit.

This process allows you to build an exponential growth model from data and use it for predictions and analysis.

Building a logarithmic model from data involves finding the best-fitting logarithmic equation that describes the relationship between the variables in your dataset. Here's a step-by-step guide with examples:

Step 1: Collect and Examine the Data

• Collect the data that you want to analyze. Ensure that the data exhibits a pattern that suggests a logarithmic relationship.

Step 2: Choose the Appropriate Logarithmic Model

• Determine whether you're dealing with a logarithmic growth or decay relationship and select the appropriate form of the logarithmic equation: $�=�\cdot \ln (�)+�$ for growth or $�=�\cdot \ln (�-�)+�$ for decay. Here, $�$, $�$, and, in the decay model, $�$, are parameters you need to determine.

Step 3: Linearize the Model (Optional)

• Depending on your data and the available statistical software, you may need to linearize the model. For logarithmic growth, the linearized equation is ${�}^{\mathrm{\prime }}=�\cdot \ln (�)+�$, and for logarithmic decay, it's ${�}^{\mathrm{\prime }}=�\cdot \ln (�-�)+�$.

Step 4: Perform Regression Analysis

• Use statistical software or spreadsheet programs to perform regression analysis on the selected model. Fit the data points to the logarithmic model. Software like Excel, Python (with libraries like NumPy and SciPy), or R can be used for this purpose.

Step 5: Interpret the Parameters

• The regression analysis will provide values for the parameters $�$, $�$, and, in the decay model, $�$. These values represent the coefficients in the logarithmic equation. Interpret their meanings in the context of your problem.

Step 6: Evaluate the Fit

• Assess the quality of the fit by examining goodness of fit statistics, such as R-squared, p-values, and residuals. Higher R-squared values indicate better fits.

Step 7: Make Predictions

• Use the estimated parameters to make predictions for values of $�$ that were not included in the original dataset. Plug these values into the logarithmic equation to estimate corresponding $�$ values.

Step 8: Plot the Fit

• Create a plot that displays the fitted logarithmic model along with the original data points. This visual representation helps you assess how well the model represents the data.

Let's go through an example of building a logarithmic growth model from data:

Example: Logarithmic Growth in Technology Adoption Suppose you have data on the adoption of a new technology in a community over time, and it follows a logarithmic growth pattern:

Years Since Introduction (x)	Adoption Rate (%) (y)
0	1
1	3
2	6
3	9
4	11

Step 1: Collect and Examine the Data The data exhibits a pattern of logarithmic growth in technology adoption.

Step 2: Choose the Appropriate Logarithmic Model We're dealing with logarithmic growth, so the appropriate model is $�=�\cdot \ln (�)+�$.

Step 3: Linearize the Model (Optional) For logarithmic growth, the linearized equation is ${�}^{\mathrm{\prime }}=�\cdot \ln (�)+�$.

Step 4: Perform Regression Analysis Perform regression analysis using software, fitting the data to the logarithmic model. The software provides estimated values for $�$ and $�$.

Step 5: Interpret the Parameters Interpret the estimated parameters $�$ and $�$ in the context of technology adoption.

Step 6: Evaluate the Fit Assess the quality of the fit using R-squared and other statistics.

Step 7: Make Predictions Use the estimated parameters to predict technology adoption rates at future time points.

Step 8: Plot the Fit Create a plot that displays the fitted logarithmic growth model and the original data points to visualize the fit.

This process allows you to build a logarithmic model from data and use it for predictions and analysis.

Performing logarithmic regression using a graphing utility typically involves the use of software or a graphing calculator. Here's a step-by-step guide on how to perform logarithmic regression with a graphing utility, using the example of a graphing calculator:

Step 1: Enter Your Data

• Turn on your graphing calculator.
• Enter your data points into the calculator. You'll usually have two lists: one for the x-values and another for the y-values. You can usually find these lists in the "STAT" or "DATA" menu on your calculator.

Step 2: Set Up the Regression

• After entering the data, go to the "STAT" or "DATA" menu, and select the option for statistical calculations.
• Choose the type of regression you want to perform. In this case, select "Logarithmic Regression" or something similar from the menu. The specific steps may vary depending on the calculator model.

Step 3: Perform the Regression

• Once you've selected logarithmic regression, the calculator will calculate the best-fitting logarithmic model for your data. It will determine the values of the parameters (a and b) in the logarithmic equation $�=�\cdot \ln (�)+�$.

Step 4: View the Results

• The calculator will display the results of the regression, including the values of the parameters (a and b) and the equation of the logarithmic model.
• Additionally, you'll likely see the R-squared value, which indicates how well the model fits the data. A higher R-squared value indicates a better fit.

Step 5: Visualize the Fit

• Plot the logarithmic model along with your original data points on the calculator's graphing screen. This visual representation helps you assess how well the model represents the data.

Step 6: Make Predictions

• You can now use the estimated values of the parameters (a and b) to make predictions for values of x that were not in the original dataset. Plug these values into the logarithmic equation to estimate corresponding y values.

Remember that the steps and specific menu options may vary depending on the brand and model of your graphing calculator. Be sure to consult your calculator's user manual or guide for precise instructions on how to perform logarithmic regression.

Building a logistic model from data involves finding the best-fitting logistic equation that describes the relationship between the variables in your dataset. The logistic model is commonly used to describe growth or diffusion processes that start slowly, accelerate, and eventually level off. Here's a step-by-step guide with examples:

Step 1: Collect and Examine the Data

• Collect the data that you want to analyze. Ensure that the data exhibits the pattern of a logistic growth or diffusion process.

Step 2: Choose the Appropriate Logistic Model

• The general form of the logistic model is $�=\frac{�}{1+{�}^{�(�-�)}}$, where $�$, $�$, and $�$ are parameters. This equation is used for logistic growth.

Step 3: Linearize the Model (Optional)

• Depending on your data and the available statistical software, you may need to linearize the model. For logistic growth, the linearized equation is ${�}^{\mathrm{\prime }}=\frac{1}{�}={�}^{-�(�-�)}$.

Step 4: Perform Regression Analysis

• Use statistical software or spreadsheet programs to perform regression analysis on the selected model. Fit the data points to the logistic model. Software like Excel, Python (with libraries like NumPy and SciPy), or R can be used for this purpose.

Step 5: Interpret the Parameters

• The regression analysis will provide values for the parameters $�$, $�$, and $�$. These values represent the coefficients in the logistic equation. Interpret their meanings in the context of your problem.

Step 6: Evaluate the Fit

• Assess the quality of the fit by examining goodness of fit statistics, such as R-squared, p-values, and residuals. Higher R-squared values indicate better fits.

Step 7: Make Predictions

• Use the estimated parameters to make predictions for values of $�$ that were not included in the original dataset. Plug these values into the logistic equation to estimate corresponding $�$ values.

Step 8: Plot the Fit

• Create a plot that displays the fitted logistic model along with the original data points. This visual representation helps you assess how well the model represents the data.

Now, let's go through an example of building a logistic growth model from data:

Example: Population Growth of an Organism Suppose you have data on the population growth of an organism over time, and it follows a logistic growth pattern:

Years (x)	Population (y)
0	100
1	200
2	500
3	800
4	950

Step 1: Collect and Examine the Data The data exhibits a pattern of logistic growth in the population of the organism.

Step 2: Choose the Appropriate Logistic Model We're dealing with logistic growth, so the appropriate model is $�=\frac{�}{1+{�}^{�(�-�)}}$.

Step 3: Linearize the Model (Optional) For logistic growth, the linearized equation is ${�}^{\mathrm{\prime }}=\frac{1}{�}={�}^{-�(�-�)}$.

Step 4: Perform Regression Analysis Perform regression analysis using software, fitting the data to the logistic model. The software provides estimated values for $�$, $�$, and $�$.

Step 5: Interpret the Parameters Interpret the estimated parameters $�$, $�$, and $�$ in the context of population growth.

Step 6: Evaluate the Fit Assess the quality of the fit using R-squared and other statistics.

Step 7: Make Predictions Use the estimated parameters to predict the population of the organism at future time points.

Step 8: Plot the Fit Create a plot that displays the fitted logistic growth model and the original data points to visualize the fit.

This process allows you to build a logistic growth model from data and use it for predictions and analysis.

College Algebra Chapter 6 Quiz

1. The population of a pod of bottlenose dolphins is modeled by the function A(t)=7(2.38)t, where t is given in years. To the nearest whole number, what will the pod population be after 4 years? 2. Find an exponential equation that passes through the points (-1, 3) and (1, 5). 3. 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? 4. An investment account was opened with an initial deposit of $8,600 and earns 6.4% interest, compounded continuously. How much will the account be worth after 19 years? 5. Graph the function f(x)=3(0.3)−x and its reflection across the y-axis on the same axes, and give the y-intercept.

Let's solve each problem step by step:

1. Population of Dolphins: The population of dolphins is modeled by the function $�(�)=7\cdot (2.38{)}^{�}$. To find the population after 4 years ($�=4$), plug in $�=4$ into the equation: $�(4)=7\cdot (2.38{)}^4$

   Use a calculator to find the numerical value.

2. Exponential Equation through Two Points: Given points (-1, 3) and (1, 5), we can use these to find the exponential equation $�=�\cdot {�}^{�}$. The general form is $�=�\cdot {�}^{�}$, and you can use the given points to create two equations and solve for $�$ and $�$.

3. Savings Goal: Drew wants to save $2,500 with a 6.25% APR compounded daily in 4 years. The compound interest formula is given by: $�=�{(1+\frac{�}{�})}^{��}$ where $�$ is the future value, $�$ is the principal (initial deposit), $�$ is the annual interest rate (as a decimal), $�$ is the number of times interest is compounded per year, and $�$ is the number of years.

   In this case, $�=2500$, $�=0.0625$ (6.25% as a decimal), $�=365$ (compounded daily), and $�=4$.

4. Continuous Compounding: For continuous compounding, the formula is $�=�\cdot {�}^{��}$, where $�$ is the base of the natural logarithm. In this case, $�=8600$, $�=0.064$ (6.4% as a decimal), and $�=19$.

5. Graphing: The function $�(�)=3\cdot (0.3{)}^{-�}$ and its reflection across the y-axis ($�(�)=-3\cdot (0.3{)}^{-�}$) can be graphed on the same axes. The y-intercept is the point where the graph intersects the y-axis ($�=0$).

7. Rewrite  log8.4(712.145)=a

  as an equivalent exponential equation.

8. Rewrite  e21=m

  as an equivalent logarithmic equation.

9. Solve for  x

  by converting the logarithmic equation  log17(x)=2

  to exponential form.

10. Evaluate  log(11,000,000)

  without using a calculator.

11. Evaluate  ln(0.516)

  using a calculator. Round to the nearest thousandth.

12. Graph the function  g(x)=log(17−5x)+3.

 

13. State the domain, vertical asymptote, and end behavior of the function  f(x)=log5(35−14x)+5.

 

14. Rewrite  log(15a⋅6b)

  as a sum.

15. Rewrite  logt(98)−logt(2)

  in compact form.

16. Rewrite  log8(a1b)

  as a product.

17. Use properties of logarithm to expand  ln(y3z2⋅x−4−−−−√3).

 

18. Condense the expression  6ln(c)+ln(d)+ln(a)3+ln(b+3)3

  to a single logarithm.

19. Rewrite  163x−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.

20. Solve  (181)x⋅1243=(19)−3x−1

  by rewriting each side with a common base.

21. Use logarithms to find the exact solution for  −9e10a−8−5=−41

  . If there is no solution, write no solution.

22. Find the exact solution for  10e4x+2+5=66.

  If there is no solution, write no solution.

23. Find the exact solution for  −5e−4x−1−4=64.

  If there is no solution, write no solution.

24. Find the exact solution for  2x−3=62x−1.

  If there is no solution, write no solution.

25. Find the exact solution for  e2x−ex−72=0.

  If there is no solution, write no solution.

26. Use the definition of a logarithm to find the exact solution for  4log(2n)−7=−11

 

27. Use the one-to-one property of logarithms to find an exact solution for  log(4x2−10)+log(3)=log(51)

  If there is no solution, write no solution.

28. The formula for measuring sound intensity in decibels  D

  is defined by the equation  D=12log(II0),

  where  I

  is the intensity of the sound in watts per square meter and  I0=14−15

  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  3.4⋅20−1

  watts per square meter?

29. A radiation safety officer is working with  182

  grams of a radioactive substance. After  27

  days, the sample has decayed to  70

  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?

30. Write the formula found in the previous exercise as an equivalent equation with base  e.

  Express the exponent to five significant digits.

31. A bottle of soda with a temperature of  70°

  Fahrenheit was taken off a shelf and placed in a refrigerator with an internal temperature of  32° F.

  After ten minutes, the internal temperature of the soda was  68° F.

  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?

32. The population of a wildlife habitat is modeled by the equation  P(t)=2501+5.2e−0.33t,

  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?

Let's work through these problems one by one:

7. Rewrite as an Exponential Equation: ${\log }_{8.4}(712.145)=�$ can be rewritten as $8.4^{�}=712.145$.

8. Rewrite as a Logarithmic Equation: ${�}^{21}=�$ can be rewritten as $\ln (�)=21$.

9. Solve for x by Converting to Exponential Form: ${\log }_{17}(�)=2$ can be rewritten as $17^2=�$. Solve for $�$.

10. Evaluate log(11,000,000) Without a Calculator: The base 10 logarithm of 11,000,000 is equal to 7 because $10^7=10,000,000$, and $10,000,000<11,000,000<100,000,000$.

11. Evaluate ln(0.516): Using a calculator, $\ln (0.516)\approx -0.659$.

12. Graph g(x): Graph the function $�(�)=\log (17-5�)+3$.

13. State the Domain, Vertical Asymptote, and End Behavior: The domain of $�(�)={\log }_5(35-14�)+5$ is the set of all real numbers where the argument of the logarithm is greater than zero (i.e., $35-14�>0$). To find the vertical asymptote, set the argument equal to zero and solve for $�$. The end behavior depends on the sign of the coefficient of the logarithm.

14. Rewrite log(15a * 6b) as a Sum: $\log (15�\cdot 6�)=\log (15�)+\log (6�)$.

15. Rewrite logt(98) - logt(2) in Compact Form: ${\log }_{�}(98)-{\log }_{�}(2)={\log }_{�}\left(\frac{98}{2}\right)={\log }_{�}(49)$.

16. Rewrite log8(a^b) as a Product: ${\log }_8({�}^{�})=�\cdot {\log }_8(�)$.

17. Expand ln(y^3z^2 * x^(-4/3)): $\ln ({�}^3{�}^2\cdot {�}^{-4\mathrm{/}3})=3\ln (�)+2\ln (�)-\frac{4}{3}\ln (�)$.

18. Condense the Expression: $6\ln (�)+\ln (�)+\ln ({�}^3)+\ln ((�+3{)}^3)=\ln ({�}^6)+\ln (�)+\ln ({�}^3)+\ln ((�+3{)}^3)=\ln ({�}^6�({�}^3)(�+3{)}^3)\mathrm{.}$

19. Rewrite 163x - 5 = 1000 as a Logarithm: $163�-5=1000$ can be rewritten as ${\log }_{16}(1000)=�$.

20. Rewrite with a Common Base: Rewrite both sides with a common base, e.g., base 10 or base 2, and then solve for $�$.

21. Solve -9e^(10a - 8) - 5 = -41: Solve for $�$ by isolating ${�}^{10�-8}$ on one side and then taking the natural logarithm of both sides.

22. Solve 10e^(4x + 2) + 5 = 66: Solve for $�$ using a similar method as in the previous problem.

23. Solve -5e^(-4x - 1) - 4 = 64: Isolate ${�}^{(}-4�-1)$ on one side and then solve for $�$.

24. Solve 2x - 3 = 62x - 1: Rearrange the equation to isolate the exponential term and then solve for $�$.

25. Solve e^(2x) - e^x - 72 = 0: Set ${�}^{�}=�$ and solve a quadratic equation in $�$.

26. Solve 4log(2n) - 7 = -11: Isolate $4\log (2�)$ on one side and then solve for $\log (2�)$. Finally, convert to exponential form.

27. Solve log(4x^2 - 10) + log(3) = log(51): Combine the logarithms and then solve for $�$.

28. Find the Number of Decibels: Use the given formula to find the number of decibels from the provided sound intensity.

29. Radioactive Decay Half-Life: Write an exponential equation for the decay and then solve for the half-life.

30. Express the Half-Life Equation with Base e: Rewrite the equation with base $�$ to find the exponent to five significant digits.

31. Newton’s Law of Cooling: Use Newton's Law of Cooling to model the temperature of the soda after one hour.

32. Wildlife Habitat Population: Solve for the original number of animals transported to the habitat and the time to reach half the capacity.