# Exponential and Logarithmic Functions

## Fitting Exponential Models to Data


### Learning Objectives

1. Draw and interpret scatter diagrams (linear, exponential, logarithmic). (CA 4.3.1)
2. Fit a regression equation to a set of data and use the linear (or exponential) model to make predictions. (CA 4.3.4)

### Objective 1: Draw and interpret scatter diagrams (linear, exponential, logarithmic). (CA 4.3.1)

A Scatter Plot is a graph of plotted points that may show a relationship between the variables in a set of data.

### Practice Makes Perfect

Draw and interpret scatter diagrams ( linear, exponential, logarithmic).

### Objective 2: Fit a regression equation to a set of data and use the linear (or exponential) model to make predictions. (CA 4.3.4)


We can find a linear function that fits the data in the previous problem by “eyeballing” a line that seems to fit. But while estimating a line works relatively well, technology can help us find a line that fits the data as perfect as possible.

This line is called the Least Squares Regression Line or Linear Regression Model.
A regression line is a line that is closest to the data in the scatter plot, which means that such a line is a best fit for the data.
Fit a regression equation to a set of data and use the linear (or exponential) model to make predictions.

### Practice Makes Perfect

Fit a regression equation to a set of data and use the linear (or exponential) model to make predictions.
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 
  or 
  When performing regression analysis, we use the form most commonly used on graphing utilities, 
  Take a moment to reflect on the characteristics we’ve already learned about the exponential function 
  (assume 
  
1.  
  must be greater than zero and not equal to one.
2. The initial value of the model is 
 

As part of the results, your calculator will display a number known as the correlation coefficient, labeled by the variable 
  or 
  (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 
  instead of 
  but the closer either value is to 1, the better the regression equation approximates the data.

### Building a Logarithmic Model from Data

Just as with exponential functions, there are many real-world applications for logarithmic functions: intensity of sound, pH levels of solutions, yields of chemical reactions, production of goods, and growth of infants. As with exponential models, data modeled by logarithmic functions are either always increasing or always decreasing as time moves forward. Again, it is the way they increase or decrease that helps us determine whether a logarithmic model is best.
Recall that logarithmic functions increase or decrease rapidly at first, but then steadily slow as time moves on. By reflecting on the characteristics we’ve already learned about this function, we can better analyze real world situations that reflect this type of growth or decay. When performing logarithmic regression analysis, we use the form of the logarithmic function most commonly used on graphing utilities, 
  For this function
1. All input values, 
  must be greater than zero.
2. The point 
  is on the graph of the model.
3. If 
  the model is increasing. Growth increases rapidly at first and then steadily slows over time.
4. If 
  the model is decreasing. Decay occurs rapidly at first and then steadily slows over time.

### Building a Logistic Model from Data

Like exponential and logarithmic growth, logistic growth increases over time. One of the most notable differences with logistic growth models is that, at a certain point, growth steadily slows and the function approaches an upper bound, or limiting value. Because of this, logistic regression is best for modeling phenomena where there are limits in expansion, such as availability of living space or nutrients.
It is worth pointing out that logistic functions actually model resource-limited exponential growth. There are many examples of this type of growth in real-world situations, including population growth and spread of disease, rumors, and even stains in fabric. When performing logistic regression analysis, we use the form most commonly used on graphing utilities:
Recall that:
1.  
  is the initial value of the model.
2. when 
  the model increases rapidly at first until it reaches its point of maximum growth rate, 
  At that point, growth steadily slows and the function becomes asymptotic to the upper bound 
  
3.  
 
is the limiting value, sometimes called the carrying capacity, of the model.

### Key Concepts

1. Exponential regression is used to model situations where 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.
2. We use the command “ExpReg” on a graphing utility to fit function of the form 
  to a set of data points. See .
3. Logarithmic regression is used to model situations where growth or decay accelerates rapidly at first and then slows over time.
4. We use the command “LnReg” on a graphing utility to fit a function of the form 
  to a set of data points. See .
5. Logistic regression is used to model situations where growth accelerates rapidly at first and then steadily slows as the function approaches an upper limit.
6. We use the command “Logistic” on a graphing utility to fit a function of the form 
  to a set of data points. See .

### Section Exercises


### Verbal


### Graphical

For the following exercises, match the given function of best fit with the appropriate scatterplot in  through . Answer using the letter beneath the matching graph.

### Numeric


### Technology

For the following exercises, use this scenario: The population 
  of a koi pond over 
  months is modeled by the function 
  
For the following exercises, use this scenario: The population 
  of an endangered species habitat for wolves is modeled by the function 
  where 
  is given in years.
For the following exercises, refer to .


For the following exercises, refer to .


For the following exercises, refer to .


For the following exercises, refer to .


For the following exercises, refer to .

For the following exercises, refer to .


### Extensions


### Chapter Review Exercises


### Exponential Functions


### Graphs of Exponential Functions


### Logarithmic Functions


### Graphs of Logarithmic Functions


### Logarithmic Properties


### Exponential and Logarithmic Equations


### Exponential and Logarithmic Models

For the following exercises, use this scenario: A doctor prescribes 
  milligrams of a therapeutic drug that decays by about 
  each hour.
For the following exercises, use this scenario: A soup with an internal temperature of 
  Fahrenheit was taken off the stove to cool in a 
  room. After fifteen minutes, the internal temperature of the soup was 
  
For the following exercises, use this scenario: The equation 
  models the number of people in a school who have heard a rumor after 
  days.
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.

### Fitting Exponential Models to Data

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.

### Practice Test

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.