Section outline

  • E.Coli bacteriaFocus 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 (Todar, PhD, Kenneth. Todar's Online Textbook of Bacteriology. http://textbookofbacteriology.net/growth_3.html).

    For simplicity’s sake, suppose we begin with a culture of one bacterial cell that can divide every hour. Table 1 below shows the number of bacterial cells at the end of each subsequent hour. We would 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!

    Table 1
    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.

    Image Caption: Electron micrograph of E.Coli bacteria (credit: “Mattosaurus,” Wikimedia Commons)

    (Content & Image Source: Chapter 6 Introduction, Algebra and Trigonometry 2e, Jay Abramson, OpenStax, CC BY 4.0 License)

    Upon completion of this module, you will be able to:

    Section 6.1 Exponential Functions
    • Evaluate exponential functions.
    • Find the equation of an exponential function.
    • Use compound interest formulas.
    • Evaluating Functions with Base e.

    Section 6.2 Graphs of Exponential Functions
    • Graph exponential functions.
    • Graph exponential functions using transformations.

    Section 6.3 Logarithmic Functions
    • Convert from logarithmic to exponential form.
    • Convert from exponential to logarithmic form.
    • Evaluate logarithms.
    • Use common logarithms.
    • Use natural logarithms.

    Section 6.4 Graphs of Logarithmic Functions
    • Identify the domain of a logarithmic function.
    • Graph logarithmic functions.
    • Graphing Transformations of Logarithmic Functions.

    Section 6.5 Logarithmic Properties
    • Use the product rule for logarithms.
    • Use the quotient rule for logarithms.
    • Use the power rule for logarithms.
    • Expand logarithmic expressions.
    • Condense logarithmic expressions.
    • Use the change-of-base formula for logarithms.

    Section 6.6 Exponential and Logarithmic Equations
    • Use like bases to solve exponential equations.
    • Use logarithms to solve exponential equations.
    • Use the definition of a logarithm to solve logarithmic equations.
    • Use the one-to-one property of logarithms to solve logarithmic equations.
    • Solve applied problems involving exponential and logarithmic equations.

    Section 6.7 Exponential and Logarithmic Models
    • Model exponential growth and decay.
    • Use Newton’s Law of Cooling.
    • Use logistic-growth models.
    • Choose an appropriate model for data.
    • Express an exponential model in base e.

    To achieve these objectives:
    1. Read the Module 6 Introduction (see above).
    2. Read Sections 6.1 - 6.7 of Chapter 6: Exponential and Logarithmic Functions in College Algebra (links to each Section provided below)
      • At the end of each Section there is a list of Key Concepts, Section Exercises, and a Glossary of terms
    3. Complete the MyOpenMath Homework Assignments for each Section (links provided below) - These are graded!
    4. Practice the problems on the Chapter 6 Review Exercises, checking the solutions provided (link provided below)
    5. View the Chapter 6 Practice Test (link provided below)
    6. Complete the MyOpenMath Quiz for Chapter 6 (link provided below) - This is graded!
    7. Once you complete the Quiz, upload your work in the Quiz Work Upload Assignment using the submission link below.
    8. Post in the Chapter 6 Q&A Discussion Forum - link provided below.

    Note the check boxes to the right that help you track your progress: some are automatic, and some are manual.

    Module Pressbooks Resources and Activities

    You will find the following resources and activities in this module at the Pressbooks website. Click on the links below to access or complete each item.

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