# MA002: Precalculus I

Unit 4: Exponential and Logarithmic Functions   Imagine that a patient is given 3 mg of a medication after an operation. How long before the level of medication reduces to ½ mg? Is the medication ever truly gone out of the patient’s system? The previous functions studied have entailed a variable raised to a numeric power. This unit will introduce two new functional types which reverse this relationship and investigate a number raised to a variable power. These exponential and logarithmic functions are amazing tools for analyzing many real-world situations.

This last unit will introduce exponential functions and their inverse functions, logarithms. You will explore the graphs, domains and ranges, and applications of these functions. Once logarithms and their properties are learned, you will be able to solve exponential and logarithmic equations which are needed to analyze situations such as population growth, radioactive decay, acidity, investments, and logistic growth. Furthermore, you will investigate technology with regression activities involving linearizing exponential functions which is used in statistical analysis.

This unit should take approximately 25.5 hours to complete.

☐    Subunit 4.1: 4.25 hours

☐    Subunit 4.2: 2.75 hours

☐    Subunit 4.3: 3.75 hours

☐    Subunit 4.4: 2.75 hours

☐    Subunit 4.5: 2.75 hours

☐    Subunit 4.6: 3.75 hours

☐    Subunit 4.7: 3 hours

☐    Review and Exam: 2.5 hours

Unit4 Learning Outcomes
Upon successful completion of this unit, you will be able to: - graph exponential and logarithmic functions analytically and using technology; - computer the domain and range of exponential and logarithmic functions; - create equations of exponential and logarithmic functions; - apply the properties of logarithms to expand and condense logarithmic expressions; - solve exponential and logarithmic equations; - identify an appropriate approach, solve exponential and logarithmic application problems, and then interpret the solution; and - utilize technology to model exponential and logarithmic data and linearize exponential data.

4.1 Exponential Functions   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 4: Exponential and Logarithmic Functions” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 4: Exponential and Logarithmic Functions” (PDF)

Instructions: Read pages 215–226 of Chapter 4 to learn about exponential functions and their graphs, logarithmic functions and their graphs, properties of logarithms, real-world modeling, and exponential regression. Note that this reading covers the material you need to know for subunits 4.1.1 through 4.1.4.

Reading this section should take approximately 45 minutes.

4.1.1 Defining Exponential Functions and Properties   Note: This subunit is covered in the reading assigned beneath subunit 4.1. To learn how exponential functions model geometric change, read the information on pages 215 and 216.

Instructions: Watch this video to get an introduction of what exponential functions are and what their graphs reveal about their properties. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 30 minutes.

4.1.2 Finding Equations of Exponential Functions   Note: This subunit is covered in the reading assigned beneath subunit 4.1. To learn how to find equations to represent compound interest or radioactive decay through a growth rate or a pair of coordinates, read the information on pages 217 through 221.

• Web Media: YouTube: Mathispower4u: “Find an Exponential Function Given Two Points – Initial Value Not Given” Link: YouTube: Mathispower4u: “Find an Exponential Function Given Two Points – Initial Value Not Given” (YouTube)

Instructions: Watch this video to find the equation of an exponential function given two points that the function goes through. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 15 minutes.

4.1.3 Compound Interest   Note: This subunit is covered in the reading assigned beneath subunit 4.1. To learn how to compute the compound interest on a bank account or compute the annual percentage yield to compare two different accounts, read the information on pages 222 through 224.

Instructions: Watch this video to learn the terms related to the compound interest formula and work out real-world problems related to interest. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 15 minutes.

4.1.4 Continuous Growth Equation   Note: This subunit is covered in the reading assigned beneath subunit 4.1. To learn a new formula for compounding every second of every day and Euler’s Number, read the information on pages 224 through 226.

• Web Media: YouTube: Mathispower4u: “Exponential Function Application (y=aekt) – Bacteria Growth” Link: YouTube: Mathispower4u: “Exponential Function Application (y=aekt) – Bacteria Growth” (YouTube)

Instructions: Watch this video to discover how the continuous growth function can be used in a specific situation involving the growth rate of bacteria. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 10 minutes.

• Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 4.1: Exponential Functions – Graded Homework” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 4.1: Exponential Functions – Graded Homework” (HTML)

Instructions: Once you have completed the readings and viewed the material for all of subunit 4.1, complete this activity. You will need to create a free account in order to access the material. If you have already created an account, log in to access the activity. Your answers will be scored upon submission.

Completing this assessment should take approximately 1 hour.

• Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Quiz 10a: Section 4.1” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Quiz 10a: Section 4.1” (PDF)

Instructions: Once you have completed the readings and viewed the material for all of subunit 4.1, complete this quiz.

Once you have completed this assessment, you may check your answers against the Saylor Foundation’s “Quiz 10a: Section 4.1 Answer Key” (PDF).

Completing this assessment should take approximately 30 minutes.

4.2 Graphs of Exponential Functions   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 4: Exponential and Logarithmic Functions” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 4: Exponential and Logarithmic Functions” (PDF)

Instructions: Read pages 233–238 of Chapter 4 to learn about graphs of exponential functions, including transformations. Note that this reading covers the material you need to know for subunits 4.2.1 through 4.2.2.

Reading this section should take approximately 15 minutes.

4.2.1 Graphical Features of Exponential Functions   Note: This subunit is covered in the reading assigned beneath subunit 4.2. To learn about characteristics of exponential functions such as their vertical intercept, whether they increase or decrease, their domain and range, and their long-term behavior, read the information on pages 232 through 235.

• Web Media: YouTube: University of Houston-Downtown, Computer and Mathematical Sciences Department: “Graphs of Exponential Functions” Link: YouTube: University of Houston-Downtown, Computer and Mathematical Sciences Department: “Graphs of Exponential Functions” (YouTube)

Instructions: Watch this video, which graphs two exponential functions, noting the explanations of why a function may increase or decrease. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 10 minutes.

4.2.2 Transformations of Exponential Functions   Note: This subunit is covered in the reading assigned beneath subunit 4.2. To see how to apply the rules for transformations to the particular features of exponential functions, read the information on pages 235 through 238.

• Web Media: YouTube: Mathispower4u: “Match the Graphs of Translated Exponential Function to Equations” Link: YouTube: Mathispower4u: “Match the Graphs of Translated Exponential Function to Equations” (YouTube)

Instructions: Watch this video, using the information you have learned from the reading, and apply it to the graphs of translated functions. Take note of why certain graphs are shifted horizontally or vertically as well as to the left or to the right. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 10 minutes.

• Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 4.2: Graphs of Exponential Functions – Graded Homework” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 4.2: Graphs of Exponential Functions – Graded Homework” (HTML)

Instructions: Once you have completed the readings and viewed the material for all of subunit 4.2, complete this assessment. You will need to create a free account in order to access the material. If you have already created an account, log in to access the activity. Your answers will be scored upon submission.

Completing this assessment should take approximately 1 hour.

• Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Quiz 11a: Section 4.2” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Quiz 11a: Section 4.2” (PDF)

Instructions: Once you have completed the readings and viewed the material for all of subunit 4.2, complete this quiz.

Once you have completed this assessment, you may check your answers against the Saylor Foundation’s “Quiz 11a: Section 4.2 Answer Key” (PDF).

Completing this assessment should take approximately 30 minutes.

4.3 Logarithmic Functions   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 4: Exponential and Logarithmic Functions” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 4: Exponential and Logarithmic Functions” (PDF)

Instructions: Read pages 242–250 of Chapter 4 to learn about logarithmic functions, how they are related to exponential functions, the common and natural log, properties of logarithms, and how to use them to solve exponential equations. Note that this reading covers the material you need to know for subunits 4.3.1 through 4.3.5.

Reading this section should take approximately 30 minutes.

4.3.1 Definition of Logarithm   Note: This subunit is covered in the reading assigned beneath subunit 4.3. To read how logarithms are the inverse functions of exponential functions, read the information on page 242.

Instructions: This video introduces logarithms as the inverse of an exponential function as well as how to convert between the two. Take detailed notes and watch the video as many times as you need. Note that this web media also covers the material you need to know for subunit 4.3.2.

Watching this video should take approximately 20 minutes.

4.3.2 Logarithm Equivalent to an Exponential   Note: This subunit is covered in the reading assigned beneath subunit 4.3. To understand and practice how each exponential equation can be written as a logarithm, read the information on pages 243 and 244. This subunit is also covered in the web media assigned beneath subunit 4.3.1.

4.3.3 Common and Natural Logarithm   Note: This subunit is covered in the reading assigned beneath subunit 4.3. To learn the two most commonly used logarithms which are included on most scientific calculators, read the information on pages 244 and 245.

• Web Media: YouTube: Mathispower4u: “Write Exponential Equations as Logarithmic Equations – Natural Logarithms” Link: YouTube: Mathispower4u: “Write Exponential Equations as Logarithmic Equations – Natural Logarithms” (YouTube)

Instructions: Watch this video to see the connection between an exponential equation, logarithmic equation, and natural logarithmic equation. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 5 minutes.

4.3.4 Properties of Logarithms   Note: This subunit is covered in the reading assigned beneath subunit 4.3. To learn the exponential and change-of-base properties of logarithms, read the information on pages 245 through 247.

Instructions: Watch this video to determine the value of a logarithm using the change of base formula. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 15 minutes.

4.3.5 Solving Exponential Equations   Note: This subunit is covered in the reading assigned beneath subunit 4.3. To learn the four-step process to solve exponential equations with examples involving pH and beetle populations, read the information on pages 247 through 250.

• Web Media: YouTube: Mathispower4u: “Solving Exponential Equations – Part 2 of 2” Link: YouTube: Mathispower4u: “Solving Exponential Equations – Part 2 of 2” (YouTube)

Instructions: Watch this video, which covers the four steps to solving an exponential equation using logarithms. This video also explains how to check your work graphically. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 20 minutes.

• Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 4.3: Logarithmic Functions – Graded Homework” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 4.3: Logarithmic Functions – Graded Homework” (HTML)

Instructions: Once you have completed the readings and viewed the material for all of subunit 4.3, complete this activity. You will need to create a free account in order to access the material. If you have already created an account, log in to access the activity. Your answers will be scored upon submission.

Completing this assessment should take approximately 1 hour.

• Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Quiz 12a: Section 4.3” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Quiz 12a: Section 4.3” (PDF)

Instructions: Once you have completed the readings and viewed the material for all of subunit 4.3, complete this quiz.

Once you have completed this assessment, you may check your answers against the Saylor Foundation’s “Quiz 12a: Section 4.3 Answer Key” (PDF).

Completing this assessment should take approximately 30 minutes.

4.4 Logarithmic Properties   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 4: Exponential and Logarithmic Functions” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 4: Exponential and Logarithmic Functions” (PDF)

Instructions: Read pages 253–259 of Chapter 4 to learn additional properties of logarithms to aid in solving logarithmic equations and more complex exponential equations. Note that this reading covers the material you need to know for subunits 4.4.1 through 4.4.2.

Reading this section should take approximately 30 minutes.

4.4.1 Properties of Logarithms   Note: This subunit is covered in the reading assigned beneath subunit 4.4. To learn the sum and difference properties of logs and review the others learned in Chapter 3, read the information on pages 253 through 256.

• Web Media: YouTube: Mathispower4u: “Combine a Sum and Difference of Two Logarithms” Link: YouTube: Mathispower4u: “Combine a Sum and Difference of Two Logarithms” (YouTube)

Instructions: Watch this video to see an example of how to use the sum and difference properties to simplify a logarithmic equation. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 10 minutes.

4.4.2 Log Properties in Solving Equations   Note: This subunit is covered in the reading assigned beneath subunit 4.4. To learn how to solve logarithmic and exponential equations, which is essential in the sciences, read the information on pages 256 through 259.

Instructions: Watch this video, which demonstrates how to use the properties of exponential and logarithmic functions to solve equations. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 15 minutes.

• Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 4.4: Log Properties – Graded Homework” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 4.4: Log Properties – Graded Homework” (HTML)

Instructions: Once you have completed the readings and viewed the material for all of subunit 4.4, complete this assessment. You will need to create a free account in order to access the material. If you have already created an account, log in to access the activity. Your answers will be scored upon submission.

Completing this assessment should take approximately 1 hour.

4.5 Graphs of Logarithmic Functions   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 4: Exponential and Logarithmic Functions” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 4: Exponential and Logarithmic Functions” (PDF)

Instructions: Read pages 262–266 of Chapter 4 to learn about graphs of logarithmic functions of different bases and how to apply basic transformation techniques. Note that this reading covers the material you need to know for subunits 4.5.1 through 4.5.2.

Reading this section should take approximately 15 minutes.

4.5.1 Graphical Features of Logarithms   Note: This subunit is covered in the reading assigned beneath subunit 4.5. To learn the basic graphical features of logarithms such as the vertical asymptote, domain, range, and long-term behavior, read the information on pages 262 and 263.

• Web Media: YouTube: Mathispower4u: “Vertical Asymptotes and Domain of Logarithmic Functions” Link: YouTube: Mathispower4u: “Vertical Asymptotes and Domain of Logarithmic Functions” (YouTube)

Instructions: Watch this video for a more in-depth example of how to find the asymptotes and domain of a logarithmic function in order to graph it. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 10 minutes.

4.5.2 Transformations of Logarithmic Functions   Note: This subunit is covered in the reading assigned beneath subunit 4.5. To see how basic transformations shift and reflect logarithmic functions, read the information on pages 264 through 266.

Instructions: Watch this video to see the graphs of logarithmic transformations. Pay attention to the detailed description of how to use the graphical features discovered in section 4.5.1 to graph and compare these functions. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 20 minutes.

• Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 4.5: Graphs of Log Functions – Graded Homework” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 4.5: Graphs of Log Functions – Graded Homework” (HTML)

Instructions: Once you have completed the readings and viewed the material for all of subunit 4.5, complete this assessment. You will need to create a free account in order to access the material. If you have already created an account, log in to access the activity. Your answers will be scored upon submission.

Completing this assessment should take approximately 1 hour.

• Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Quiz 13a: Section 4.4–4.5” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Quiz 13a: Section 4.4–4.5” (PDF)

Instructions: Once you have completed the readings and viewed the material for all of subunits 4.4 and 4.5, complete this quiz.

Once you have completed this assessment, you may check your answers against the Saylor Foundation’s “Quiz 13a: Section 4.4-4.5 Answer Key” (PDF).

Completing this assessment should take approximately 30 minutes.

4.6 Exponential and Logarithmic Models   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 4: Exponential and Logarithmic Functions” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 4: Exponential and Logarithmic Functions” (PDF)

Instructions: Read pages 270–281 of Chapter 4 to learn about exponential and logarithmic models for radioactive isotopes, exponential growth, Newton’s Law of Cooling, and logarithmic scales and how they apply to earthquakes. Note that this reading covers the material you need to know for subunits 4.6.1 through 4.6.5.

Reading this section should take approximately 45 minutes.

4.6.1 Radioactive Decay and Half-Life   Note: This subunit is covered in the reading assigned beneath subunit 4.6. To learn what the half-life is for a radioactive isotope and how it can be used to determine the age of historical artifacts, read the information on pages 270 through 272.

Instructions: Watch this video, which explains how to use the exponential decay model and determine the half-life of a sample. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 20 minutes.

4.6.2 Doubling Time   Note: This subunit is covered in the reading assigned beneath subunit 4.6. To learn how long it will take for the number of cancerous cells to double or how many people will visit a website in a year, read the information on pages 273 and 274.

• Web Media: YouTube: Mathispower4u: “Exponential Growth App (y=abt) – Given Doubling Time” Link: YouTube: Mathispower4u: “Exponential Growth App (y=abt) – Given Doubling Time” (YouTube)

Instructions: Watch this video to learn about exponential growth and apply it to a study on bacteria. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 10 minutes.

4.6.3 Newton’s Law of Cooling   Note: This subunit is covered in the reading assigned beneath subunit 4.6. To learn how to use Newton’s Law of Cooling (and heating) to determine how long it will take for hot items to cool to a particular temperature or cool items to warm if placed in a warm room, read the information on pages 274 through 276.

• Web Media: YouTube: Mathispower4u: “Newton’s Law of Cooling – Exponential Function App” Link: YouTube: Mathispower4u: “Newton’s Law of Cooling – Exponential Function App” (YouTube)

Instructions: Watch this video to apply Newton’s Law of Cooling to a real-world cooking situation. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 15 minutes.

4.6.4 Logarithmic Scales   Note: This subunit is covered in the reading assigned beneath subunit 4.6. To learn how logarithms are used to reduce the magnitude of sets of very large numbers to make them easier to compute with and display, read the information on pages 276 through 278.

Instructions: Watch this video to see a thorough explanation of the difference between the linear and logarithmic scale. In addition, note how the logarithmic scale can be useful in specific situations using order of magnitude. Take detailed notes and watch the video as many times as you need. Note that this web media also covers the material you need to know for subunit 4.6.5.

Watching this video should take approximately 25 minutes.

4.6.5 Magnitude   Note: This subunit is covered in the reading assigned beneath subunit 4.6. To learn the logarithmic formula for the difference in orders of magnitude and its applications to earthquakes, read the information on pages 279 through 281. This subunit is also covered in the web media assigned beneath subunit 4.6.4.

• Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 4.6: Modeling – Graded Homework” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 4.6: Modeling – Graded Homework” (HTML)

Instructions: Once you have completed the readings and viewed the material for all of subunit 4.6, complete this assessment. You will need to create a free account in order to access the material. If you have already created an account, log in to access the activity. Your answers will be scored upon submission.

Completing this assessment should take approximately 1 hour.

4.7 Fitting Exponentials to Data   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 4: Exponential and Logarithmic Functions” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 4: Exponential and Logarithmic Functions” (PDF)

Instructions: Read pages 289–294 of Chapter 4 to learn about converting exponential data to linear data using semi-log and log-log scales. Note that this reading covers the material you need to know for subunits 4.7.1 through 4.7.3.

Reading this section should take approximately 15 minutes.

4.7.1 Semi-Log and Log-Log Graphs   Note: This subunit is covered in the reading assigned beneath subunit 4.7. To learn how to convert one or both axes to a logarithmic scale to transform an exponential curve to a linear function, read the information on pages 289 through 292.

• Web Media: YouTube: Mathispower4u: “Find an Exponential Function for a Semi-Log Graph” Link: YouTube: Mathispower4u: “Find an Exponential Function for a Semi-Log Graph” (YouTube)

Instructions: This video explains how to apply the knowledge of logarithmic scale in a graph to find an exponential function. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 10 minutes.

4.7.2 Fitting Exponential Functions to Data using Linearization   Note: This subunit is covered in the reading assigned beneath subunit 4.7. To learn how to find an exponential function for data by first using linearization and then converting it to an exponential function, read the information on pages 292 through 294.

• Web Media: YouTube: Mathispower4u: “Determine if a Table Represents a Linear or Exponential Function” Link: YouTube: Mathispower4u: “Determine if a Table Represents a Linear or Exponential Function” (YouTube)

Instructions: Watch this video to determine how to use a table to establish whether a function can be represented linearly or exponentially. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 10 minutes.

• Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 4.7: Exponential Regression – Graded Homework” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 4.7: Exponential Regression – Graded Homework” (HTML)

Instructions: Once you have completed the readings and viewed the material for all of subunit 4.7, complete this assessment. You will need to create a free account in order to access the material. If you have already created an account, log in to access the activity. Your answers will be scored upon submission.

Completing this assessment should take approximately 1 hour.

• Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Quiz 14: Sections 4.1–4.5” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Quiz 14: Sections 4.1–4.5” (PDF)

Instructions: Once you have completed the readings and viewed the material for all of subunits 4.6 and 4.7, complete this quiz.

Once you have completed this assessment, you may check your answers against the Saylor Foundation’s “Quiz 14: Section 4.1-4.5 Answer Key” (PDF).

Completing this assessment should take approximately 30 minutes.

Unit 4 Review   - Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Chapter 4 Review Problems” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Chapter 4 Review Problems” (HTML)

`````` Instructions: Once you have completed the readings and viewed the
material for all of unit 4, complete this assessment. You will need
provide a review for your unit assessment.

Completing this review should take approximately 1 hour and 30
minutes.

`````` Completing this exam should take approximately 1 hour.