Subject:
Mathematics
Material Type:
Module
Provider:
Ohio Open Ed Collaborative
Tags:
Algebra, College Algebra, Exponential Functions, Logarithmic Functions, Modeling, Tmm0012
Language:
English
Media Formats:

# Exponential and Logarithmic Functions ## Lesson Overview

This material covers Chapter 6: Exponential and Logarithmic Funcitons chapter of the OpenStax College Algebra Text. This module contains an overview of learning objectives mapped to the OTM state standards, worksheets that correspond to chapter sections, interactive Desmos Activities that pair with the chapter, and a list of supplemental videos that correspond to the chapter content.

Image by Arek Socha from Pixabay

# Exponential and Logarithimic Functions

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.16

For simplicity’s sake, suppose we begin with a culture of one bacterial cell that can divide every hour. Table 1 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!

 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 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.

(From OpenStax College Algebra)

Chapter Sections

• 6.1 Exponential Functions

• 6.2 Graphs of Exponential Functions

• 6.3 Logarithmic Functions

• 6.4 Graphs of Logarithmic Functions

• 6.5 Logarithmic Properties

• 6.6 Exponential and Logarithmic Equations

• 6.7 Exponential and Logarithmic Models

• 6.8 Fitting Exponential Models to Data

1a.Analyze functions. Routine analysis includes discussion of domain, range, zeros, general function behavior (increasing, decreasing, extrema, etc.).In addition to performing rote processes, the student can articulate reasons for choosing a particular process, recognize function families and anticipate behavior, and explain the implementation of a process (e.g., why certain real numbers are excluded from the domain of a given function).*

1b.Convert between different representations of a function.

1c.Perform operations with functions including addition, subtraction, multiplication, division, composition, and inversion; connect properties of constituent functions to properties of the resultant function; and resolve a function into a sum, difference, product, quotient, and/or composite of functions.

2.Equations and Inequalities:Successful College Algebra students are proficient at solving a wide array of equations and inequalities involving linear, quadratic, higher-order polynomial, rational, exponential, logarithmic, radical, and piecewise-defined functions (including absolute value).

2a.Recognize function families as they appear in equations and inequalities and choose an appropriate solution methodology for a particular equation or inequality and can communicate reasons for that choice.

2b.Use correct, consistent, and coherent notation throughout the solution process to a given equation or inequality.

2c.Distinguish between exact and approximate solutions and which solution methodologies result in which kind of solutions.

2d.Demonstrate an understanding of the correspondence between the solution to an equation, the zero of a function, and the point of intersection of two curves.*

3.Equivalencies:Successful College Algebra students are proficient in creating equivalencies in order to simplify expressions, solve equations and inequalities, or take advantage of a common structure or form.

3a.Purposefully create equivalences and indicate where they are valid.

3b.Recognize opportunities to create equivalencies in order to simplify workflow.

4.Modeling with Functions:Successful College Algebra students should have experience in using and creating mathematics which model a wide range of phenomena.

4a.Interpret the function correspondence and behavior of a given model in terms of the context of the model.

4c.Determine parameters of a model given the form of the model and data.

5.Appropriate Use of Technology: Successful College Algebra students are proficient at choosing and applying technology to assist in analyzing functions.

5b.Use technology to verify solutions to equations and inequalities obtained algebraically.

5c.Use technology to obtain solutions to equations to equations and inequalities which are difficult to obtain algebraically and know the difference between approximate and exact solutions.

5d.Use technology and algebra in concert to locate and identify exact solutions.

Chapter Section Objectives and related OTM standards:

6.1 Exponential Functions

• Evaluate exponential functions.

• Find the equation of an exponential function.

• Use compound interest formulas.

• Evaluate exponential functions with base e

6.2 Graphs of Exponential Functions

• Graph exponential functions. (1a, 1b, 4a)

• Graph exponential functions using transformations. (1a, 1b, 4a)

6.3 Logarithmic Functions

• Convert from logarithmic to exponential form. (1b)

• Convert from exponential to logarithmic form. (1b)

• Evaluate logarithms.

• Use common logarithms.

• Use natural logarithms.

6.4 Graphs of Logarithmic Functions

• Identify the domain of a logarithmic function. (1b, 4a, 4b)

• Graph logarithmic functions. (1b, 4a)

6.5 Logarithmic Properties

• Use the product rule for logarithms. (3a)

• Use the quotient rule for logarithms. (3a)

• Use the power rule for logarithms. (3a)

• Expand logarithmic expressions. (3a)

• Condense logarithmic expressions. (3a, 3b)

• Use the change-of-base formula for logarithms (3a, 5c)

6.6 Exponential and Logarithmic Equations

• Use like bases to solve exponential equations. (2a, 2b, 2c, 3a, 5c)

• Use logarithms to solve exponential equations. (2a, 2c, 3a, 5c)

• Use the definition of a logarithm to solve logarithmic equations. (2a, 2c, 3a, 3b, 5c)

• Use the one-to-one property of logarithms to solve logarithmic equations. (2a, 2c, 3a, 5c)

• Solve applied problems involving exponential and logarithmic equations. (2a, 2c, 3a, 5b, 5c)

6.7 Exponential and Logarithmic Models

• Model exponential growth and decay. (2a, 2c, 3a, 5b, 5c)

• Use Newton’s Law of Cooling. (2a, 2c, 3a, 5b, 5c)

• Use logistic-growth models. (2a, 2c, 3a, 4a, 4b,4c, 4d, 5b, 5c)

• Choose an appropriate model for data. (2a, 2c, 3a, 4a, 4b, 4c 4d, 5b, 5c)

• Express an exponential model in base e  (2a, 2c, 3a, 5c)

# Section 2: Chapter 6 Worksheets

Please note that seciton 6.8 has been omitted as that content does not fall within the purview of the OTM standards for College Algebra.

The files below are worksheets created to support practice and learning for chatper 6 of the OpenStax College Algebra text.

# Section 3: Chapter 6 Desmos Activities

The links below will direct to pre-built Desmos activities. These can be easily copied, remixed, or compiled into activities for teachers to customize to their class and objectives. For more on how educators can integrate the Desmos activities into their class please click here to link to the Desmos teacher support site:

Below is a short explanation of the Desmos activates linked.

Polygraph: Exponential & Logarithmic Functions

This Custom Polygraph is designed to spark vocabulary-rich conversations about exponential and logarithmic functions. Key vocabulary that may appear in student questions includes: exponential, asymptote, logarithmic, and quadrant. In the early rounds of the game, students may notice graph features from the list above, even though they may not use those words to describe them. That’s where you can step in. After most students have played 2-3 games, consider taking a short break to discuss strategy, highlight effective questions, and encourage students in their use of increasingly precise academic language. Then ask them to play several more games, putting that precise language to work.

Mocha Modeling: Starbucks Locations

n this activity, students build a model to describe the relationship between the number of Starbucks locations in the United States and the number of years since 1992. Students then use that model to make predictions about the number of locations in 2015 and beyond. Students will also interpret the features of the graph in context. In the process, students learn that not all rapid growth is exponential growth, and that another function type (logistic) may provide a better fit when finite resources come into play.

# Exponential Bundle

The functions bundle contains a large collection of activities that are appropriate for this section. The activities that apply directly to this section are listed below:

Avi and Benita's Repair Shop
In this twist on a classic activity, students compare linear and exponential growth in the context of daily payments. One plan increases by \$100 each day, while another grows by doubling the previous day's payment.

This activity is appropriate for students who have studied linear functions but may not have an experience with exponential growth. With that in mind, it makes a great first activity in an exponential functions unit.

Polygraph: Exponentials
This Custom Polygraph is designed to spark vocabulary-rich conversations about exponentials, including how they differ from linear functions. Key vocabulary that may appear in student questions includes: increasing, decreasing, intercept, rate, asymptote, and curve. In the early rounds of the game, students may notice graph features from the list above, even though they may not use those words to describe them. That’s where you can step in. After most students have played 2-3 games, consider taking a short break to discuss strategy, highlight effective questions, and encourage students in their use of increasingly precise academic language. Then ask them to play several more games, putting that precise language to work.

What Comes Next?
In this activity, students predict "what comes next" for linear and exponential functions, based first on graphs and then on tables of values. Later, students explore connections between graphs, tables, and equations of linear and exponential functions.

Marbleslides: Exponentials
In this delightful and challenging activity, students will transform exponential functions so that the marbles go through the stars. Students will test their ideas by launching the marbles, and have a chance to revise before trying the next challenge.

Card Sort: Exponentials
In this activity, students practice what they've learned about exponential functions by matching equations to properties of the graphs they will produce. They will then use their knowledge of transforming exponential functions to pair equations with graphs.

Predicting Movie Ticket Prices
In this activity, students build a model to describe the relationship between average US movie ticket price and time. Students then use that model to make predictions about past and future ticket prices. Students also interpret the parameters of their equation in context.

Game, Set, Flat
In this activity, students will develop their understanding of the exponential relationship that describes a bouncing tennis ball. They'll learn to examine successive terms in a sequence to determine if it represents an exponential relationship or not. They'll also learn how to construct the exponential equation itself.

Two Truths and a Lie: Exponentials
Students will practice their understanding of the features and vocabulary of exponential graphs by creating an exponential curve, writing two true and one false statement about it, and inviting their peers to separate truth from lies.

Logarithm Marbleslides

In this delightful and challenging activity, students will transform exponential functions so that the marbles go through the stars. Students will test their ideas by launching the marbles, and have a chance to revise before trying the next challenge.

Basic Logarithmic Transformations

This activity explores changes of base and domain as well as horizontal shifts.  It works best after students have understanding of how a basic logarithmic graph behaves.

# Section 4: Chapter 6 Supplemental Videos

Below is a document that links to supplemental videos for this chapter. Note that these videos are not created by the publisher of the text, so some verbiage or problem solving strategies may vary from what is presented in the text.