# K12MATH011: Algebra II

Unit 7: Exponential and Logarithmic Functions   In this unit, you will be introduced to exponential and logarithmic expressions, equations, and inequalities. You will also become familiar with the connection between exponential and logarithmic functions, including the relationship between operations on exponents and logarithms. You will learn to graph exponential and logarithmic functions, convert from one base to another, and solve exponential and logarithmic equations and inequalities.

Completing this unit should take approximately 10 hours and 15 minutes.

☐     Subunit 7.1: 1 hour

☐    Subunit 7.2: 2 hours and 45 minutes

☐    Subunit 7.3: 1 hour

☐    Subunit 7.4: 1 hour and 45 minutes

☐    Subunit 7.5: 1 hour

☐    Subunit 7.6: 2 hours and 45 minutes

Unit7 Learning Outcomes
Upon successful completion of this unit, you will be able to: - Convert an equation from exponential to logarithmic form. - Graph exponential and logarithmic functions. - Convert a logarithm from one base to another. - Solve exponential and logarithmic equations and inequalities.

7.1 The Connection between Exponents and Logarithms   In short, logarithms and exponents are inverse operations. That is, if x = logny, then xn = y. This is handy information for solving one set of equations in terms of the other.

• Explanation: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 6, Section 1: Introduction to Exponential and Logarithmic Functions” Link: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 6, Section 1: Introduction to Exponential and Logarithmic Functions” (PDF)

Instructions: This chapter provides detailed information on exponential and logarithmic functions, including the connection between the two. Real-world examples of their uses are given as well. Read pages 417–432, and then try the problems at the end of the chapter. Answers are on the pages that follow.

Completing this activity should take approximately 45 minutes.

Instructions: This video discusses the connection between radical expressions and exponents.

Watching the video and writing notes should take approximately 15 minutes.

7.2 Exponential and Logarithmic Functions   These two functions are inverses of one another, meaning that if some point (a,b) is an element of one, (b,a) must be an element of the inverse.

• Explanation: CK-12: “Basic Exponential Functions” Link: CK-12: “Basic Exponential Functions” (YouTube and HTML)

Instructions: This page contains video, text, and practice problems on exponential functions and provides real-world examples of their uses in science and finance. Review this material and work both the guided practice and practice problems.

Completing this activity should take approximately 1 hour.

• Explanation: CK-12: “Logarithmic Functions” Link: CK-12: “Logarithmic Functions” (YouTube and HTML)

Instructions: This page contains detailed information on logarithmic functions and provides real-world examples of their uses in science and finance. Read the text carefully and watch the video. Then work the guided practice and practice problems.

Completing this activity should take approximately 1 hour.

• Web Media: GeoGebra: “Exponential and Logarithmic Graphs” Link: GeoGebra: “Exponential and Logarithmic Graphs” (HTML)

Instructions: This interactive graph allows you to change the base of the functions by using sliders. How does changing certain values change the graphs? Then change the bottom slider value to change the x value (input) on the exponential function. How does the x value relate to what is seen on the inverse as it changes? This proves that the log function is the inverse of the given exponential function. Note how the two correspond.

Practicing with this application and writing notes should take approximately 45 minutes.

7.3 Graphing Exponential and Logarithmic Functions   As we have seen before, inverse functions are helpful in that once we know the graph of one, we can easily find the graph of the other. Exponential and logarithmic functions are inverses, so once one is graphed, simply take the reflection across f(x) = x. That is, if (1, 5) and (-2, 8) are on the graph of a function, (5, 1) and (8, -2) are on the inverse.

Instructions: This video discusses the method for graphing a basic logarithmic function.

Watching the video and writing notes should take approximately 15 minutes.

• Web Media: GeoGebra: “Exponential vs. Logarithmic” Link: GeoGebra: “Exponential vs. Logarithmic” (HTML)

Instructions: This interactive graph allows you to change various values and see how the changes affect the graphs of exponential and logarithmic functions. The first slider changes the contraction/expansion of the graph. The second slider changes the base of the functions. The third changes the asymptotes. By default, the log function is hidden, but a check box makes it visible. Change the values to see how the graphs change. Note that there is one shift not covered here (the “inside shift”), but that is for clarity on the behavior of these inverse functions.

Practicing with this application and writing notes should take approximately 45 minutes.

7.4 Common Logarithm and Base e   The two most widely used logarithms are common (base 10) and natural (base e). Both flow from problems that have interested scientists for nearly 300 years. And because powerful calculators and computers are very recent, scientists had a need to compute large values that would otherwise take much longer to work. Logarithms, introduced by John Napier in the 1600s, are the means to doing these calculations quickly and easily. Logarithms are the basis of slide rules, which were used by engineers long before electronics made calculators feasible.

• Explanation: CK-12: “Common and Natural Logarithms” Link: CK-12: “Common and Natural Logarithms” (HTML)

Instructions: Read through the page carefully and work through the guided practice problems about common (base 10) and natural (base e) logarithms, which have been used for large calculations for a few hundred years. Answers are provided.

Completing this activity should take approximately 1 hour.

• Web Media: GeoGebra: “Transforming Natural Logarithmic Function” Link: GeoGebra: “Transforming Natural Logarithmic Function” (HTML)

Instructions: This interactive graph allows you to move, stretch, and reflect the natural logarithmic function across the x and y axes and the line y = x (to show the inverse function). There are four sliders you can use to transform the graph: (1) “a” changes the coefficient of the function, which stretches or shrinks the graph; (2) “b” changes the internal (horizontal) shift of the vertical asymptote; (3) “c” changes the external (vertical) shift of the horizontal asymptote; and (4) “n” changes the internal coefficient. There are various check boxes, but the most important is the reflection in the line y = x, as this displays the inverse of the natural log function. Change the values on the slider to see how the graph changes. Write a study guide on transformations, using the knowledge you gained on this page. In particular, note what kind of changes requires certain actions. By going backward, that is, by defining the transformation you wish and then describing the action needed to achieve it, you will acquire a solid understanding of the concept.

Practicing with this application and writing notes should take approximately 45 minutes.

7.5 Change of Base Formula   You don’t have to have a top of the line scientific calculator to compute any kind of logarithm. The change of base formula allows you to compute any general logarithm using either base 10 or natural logs.

Instructions: This video discusses the change of base formula. Note that it demonstrates how to use a calculator to achieve the desired results. After viewing the content and writing notes, write a lesson on change of base formula that you would deliver to your classmates, if you could. Explain how the change of base works and show several examples.

Watching this video, writing notes, and writing the lesson should take approximately 1 hour.

7.6 Solving Exponential and Logarithmic Functions   Logarithms and exponents are inverses. The advantage is that you may use that fact to solve one function by translating the problem to the other form very easily. Logarithms have applications in such fields as music, computing complexity, psychophysics, and forensic accounting. Exponents feature prominently in fields such as economics, biology, chemistry, physics, computer science, and data security.

• Explanation: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 6, Section 3: Exponential Equations and Inequalities” Link: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 6, Section 3: Exponential Equations and Inequalities” (PDF)

Instructions: This section contains detailed information on solving exponential equations and inequalities, including examples for the calculator. Read pages 448–457, and then try the problems at the end of the section. Answers are on the pages that follow.

Completing this activity should take approximately 45 minutes.

• Explanation: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 6, Section 4: Logarithmic Equations and Inequalities” Link: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 6, Section 4: Logarithmic Equations and Inequalities” (PDF)

Instructions: This section contains detailed information on solving logarithmic equations and inequalities, including examples for the calculator. Read pages 459–467, and then try the problems at the end of the section. Answers are on the pages that follow.

Completing this activity should take approximately 45 minutes.

Instructions: This video contains information on using the relationship between logarithms and exponents to solve logarithmic equations. It also demonstrates how to use logarithm rules (which are tied to exponent rules) to help solve them.

Watching the video and writing notes should take approximately 15 minutes.

• Checkpoint: MyOpenMath: “Beginning and Intermediate Algebra: Chapter 10.4 Practice” Link: MyOpenMath: “Beginning and Intermediate Algebra: Chapter 10.4 Practice” (HTML)

Instructions: This is an assessment on understanding exponential functions. Work all 15 problems. Answers are available for viewing by a button on each problem. There is also a link for a printed version at the bottom left of the page. Note: New versions of the 15 problems can be created by accessing the link in the upper right corner of the page. You may take a nearly unlimited number of attempts with different problems.

Completing this assessment should take approximately 30 minutes.