Unit 5: Exponential, Logarithmic, and Logistic Functions
Goal: The student will demonstrate the ability to investigate exponential, logarithmic, and logistic functions and solve real-world problems, both with and without the use of technology.




a. Sketch and analyze exponential functions and their transformations.

Mathematical Background/Clarifying Examples:
Students will use knowledge from courses and units in this course to graph transformations of exponential functions. It is important to emphasize the definition of an exponential function and its different parts for students to identify a function as a growth or decay. The y-intercept and horizontal asymptote need to be stated with graphing problems.

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Resources:
Additional Practice: This resource has students identify whether an exponential function is a growth or decay. It also has students practice graphing exponential functions as well as evaluating a half life problem.

Additional Practice: This PowerPoint contains exponential functions practice. Solutions are included on separate slides.


Website: Exponential Graph Visual: This website displays the graph of an exponential function and its transformations.
http://www.analyzemath.com/expfunction/expfunction.html

b. Define the natural base.

Mathematical Background/Clarifying Examples: Expand upon students knowledge of compound interest to define the natural base.



Mark invests $1500 in a savings account that pays 3.5% interest compounded continuously. What will be Mark's account balance after 7 years?


Answer: $1916.43



Resources:
Lesson: This PowerPoint guides students to discover the number e using the compound interests formula. Practice application problems are including using the continuous compounding formula.


Website: Self Guided Lesson: This website guides one through the origin of e and incorporates practice.
e Explanation and Activity
http://www.augustatech.edu/math/molik/numbere.pdf





c. Express the inverse of an exponential function as a logarithmic function.

Mathematical Background/Clarifying Examples:
Use concepts covered in objectives a and b to identify that a logarithmic function is an inverse of an exponential functions. Students will need rewrite an exponential function as a logarithm and vice-versa. This concept will be used in future objectives.
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Resources:

Discovery: This activity assists students in discovering the relationships between exponential and logarithmic functions.

A look back: This assignment reviews concepts of inverses. This can be a good review prior to showing that exponential and logarithmic function are inverses.



Lesson: This PowerPoint guides students through definition of logarithms and emphasizes why they are used through the use of application problems.





d. Evaluate logarithms to any base with and without a calculator.

Mathematical Background/Clarifying Examples:
Students will use knowledge of logarithms and exponential functions to evaluate with and without a calculator. Students will need to use knowledge of the zero exponent property, negative exponent property, and square roots.

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Resources:
Website: Practice: This resource reviews concepts from previous objectives and reinforces topics for evaluating logarithms. An answer key is provided.
http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Meaning%20of%20Logarithms.pdf

Practice: The resources gives students extra practice for objectives c and d.



e. Use and apply the laws of logarithms and the change of base formula.

Mathematical Background/Clarifying Examples: Guide students in recognizing the inverse relationship between logarithms and exponential functions. Use this knowledge to simplify logarithmic expressions and use in later sections to solve logarithmic and exponential equations.

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Resources:
Extra Practice: This provides practice for expanding and simplifying logarithmic expressions. Solutions are included.
http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Properties%20of%20Logarithms.pdf

Extra Practice: This provides definitions for the logarithmic properties, practice problems, and an application problem.

Extra Practice: This provides the change of base formula and practice.


Website: Calculator Practice: This resource reinforces the change of base formula using the calculator.

http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Change%20of%20Base%20Formula.pdf

Website: Interactive Lesson: This hands-on activity assists students in discovering the properties of logarithms.
http://illuminations.nctm.org/LessonDetail.aspx?id=L817


f. Sketch and analyze logarithmic functions and their transformations.

Mathematical Background/Clarifying Examples: Students will need to use their knowledge of exponential functions and graphs of inverses in order to graph logarithmic functions. Knowledge from previous units should be used to analyze transformations.

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Resources:
Website: Extra Practice: This resource provides logarithmic graphing practice. Solutions are provided.
http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Graphing%20Logarithms.pdf

Lesson: This lesson includes graphing examples and extra practice stating domain and range of a logarithmic function.


Website: Interactive Lesson: This website displays logarithmic functions and its transformations.

g. Solve exponential and logarithmic equations.

Mathematical Background/Clarifying Examples: The properties of exponential and logarithmic expressions will be used in order to solve exponential and logarithmic equations.

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Resources:
Website: Extra Practice: This resource provides practice on solving logarithmic equations. All problems require students to rewrite the equation in exponential form to solve.
http://www.augustatech.edu/math/molik/Logarithms.pdf

Lesson and Practice: This lesson has examples of solving exponential equations. An application problem is provided as well as extra practice for students.


Website: Extra Practice: The resource provides practice on solving exponential equations. All problems require students to rewrite the equation in logarithmic form.Solutions are provided.
http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Solving%20Exponential%20Equations%20with%20Logarithms.pdf

h. Sketch and analyze logistic functions.

Mathematical Background/Clarifying Examples: Guide students to use their knowledge of exponential functions to graph and analyze logistic functions. Discuss end behavior to label horizontal asymptotes of logistic functions. Word problems involving restricted growth will help students understanding of this concept.

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Resources:
Application: This problem has student write a logistic equation that models a set of data. Students are to use the graphing calculator to perform a logistic regression. There are follow up questions.


Website: This website displays logistic functions and its transformations.



i. Compare and contrast the exponential and logistic models.

Mathematical Background/Clarifying Examples: Use the definitions of both exponential and logistic functions examine similarities and differences. Emphasize restricted vs. unrestricted growth.


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Resources:
Comparing Activity: This activity has students label a table, graph, or function as exponential or logistic.




j. Solve real-world problems including exponential growth and decay, compound interest, and applications of logarithms and logistic functions.

Mathematical Background/Clarifying Examples: Use knowledge from this unit and apply its concepts to solve application problems.

Once the population of Wilde Lake High School exceeds 1800 students the school will reconsider building a new high school in the area. Currently Wilde Lake High School has 1400 student and the population is growing 4% per year. How many years will it take for the school board to consider building a new school?


Answer:
About 6.41 years




Resources:
Additional students practice: This PowerPoint contains seven word problems modeling exponential growth and decay, compound interest, and applications of logarithms and logistic functions.




Additional student practice; This resource contains application problems using logarithm and exponential functions.



Additional student practice: The resource contains practice problems using exponential, logarithmic, and logistic functions. Some problems incorporate the use of technology to identify the what type of function the data models.