Logarithmic Functions

Lesson Plan, Lab Edition

Logarithmic Functions Objectives Students will: • Graph logarithmic equations of the form y = logax. • Define logarithmic functions. • State the restrictions of logarithmic functions. • Explain the relationship between exponential and logarithmic functions. • Solve application problems involving logarithms. •

Prerequisite Knowledge Students are able to: • Graph linear, quadratic, and exponential functions. • Solve equations for a given variable. • Evaluate exponential equations involving ‘e’.

Resources •

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This lesson assumes that your classroom has enough computers for all your students, either working individually or in small groups. If your classroom has only one computer, from which you can lecture, see the lecture version of this lesson. Rulers, pencil, calculators, and paper Access to http://www.ExploreMath.com Copies of the worksheet for each student (optional)

Lesson Preparation Before conducting this lesson, be sure to read through it thoroughly, and familiarize yourself with the Logarithms activity at ExploreMath.com. Verify that the activity plays correctly on all the machines in your class, and if it doesn’t, go to http://www.exploremath.com/about/shockerhelp.cfm for troubleshooting information. You may want to bookmark the activity page for your students. If you like, make copies of the worksheet for each student.

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Lesson Motivation: Warm-up: Who wants to be a millionaire?

When someone asked Albert Einstein what he considered mankind’s greatest invention, he’s said to have replied without hesitation, "compound interest." Interest that is compounded continuously is modeled by the equation A = Pert, where ‘A’ is the amount after ‘P’ dollars is invested for ‘t’ years at rate ‘r’. Historically the U.S. stock market has an average rate of return between 8-10%. Have your students plot a graph that shows the growth of $10,000 at an interest rate of 10% over the next 10, 20, 30, 40, and 50 years. The graph should look like the one below.

$1,600,000 $1,400,000 $1,200,000 $1,000,000 $800,000 $600,000 $400,000 $200,000 $0 0

10

20

30

40

50

60

Have students notice how the graph increases. Ask students how long they would need to keep the $10,000 invested in order to have $1 million in 50 years. If the students are around 17 years old, 50 years from now will be close to the U.S. Social Security retirement age. In order to solve the problem, the equation 1,000,000 = 10000e((0.10)(t)) needs to be solved for the value of ‘t’. Have students attempt to solve the equation. Ask students what trouble they encounter when solving for ‘t’. Students should explain that they couldn’t get ‘t’ out of the exponent position. They should see that an inverse for ‘ex’ is needed to solve the equation. Tell students that they will solve this problem later in the lesson, but first they need to investigate the inverses of exponential functions. This group of inverses is called logarithms.

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Graphical inverses Have students graph an exponential function on a sheet of paper. Below is the function y=4x.

Now have students fold the paper along the line y=x. They should then trace the graph with a pencil to make an impression on the paper. They should unfold the paper and trace the impression to get the graph of the inverse function. It should look like the graph below.

Explain to students that this is the graph of the logarithmic function y = log4x. The logarithmic functions activity To explore the characteristics of logarithmic functions, have students go to the Logarithms activity at ExploreMath.com.

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Defining y = logax Have students grab the ‘a’ slide bar and move it such that a=2. This can also be accomplished by typing a 2 to the right of the slide bar.

Now have students select the “Calculate Data Values” clipboard at the bottom of the screen. Then set the minimum to 0, set the maximum to 64 and set the step to 1.

Have students pick out the x, y pairs that have integer values. These should be: x y

1 0

2 1

4 2

8 3

16 4

32 5

64 6

Remembering that the base, a, of this logarithm is 2, have students use the above numbers to try to find a way to relate a, y, and x. Lead students towards

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the idea that ay = x. Have students make conjecture about the definition of y = logax. Repeat the activity with a base of three to see if the student’s conjectures hold true. Once an informal definition of y = logax has been formed, give the students the formal definition. The logarithmic function y = logax where a > 0 and a ≠ 1 is defined by: y = logax if and only if x = ay Restrictions of logarithmic functions

Ask students to experiment with the ‘a’ slide bar to see what happens to the graph when ‘x’ approaches the y-axis. Ask students if the graph ever crosses the y-axis. Ask students to conjecture why the graphs never enter the second or third quadrants. Have students look to the definition for help in making their conjectures. Students could substitute a negative value for ‘x’ in the definition and see what happens. y = log2-4 if and only if -4 = 2y Students should see that the definition breaks down when ‘x’ is negative because there exist no real exponent that when combined with a positive base yields a negative answer. Ask students why the condition a > 0 is included in the definition. Have students select the ‘show related exponential’ and ‘show line y=x’ boxes.

Remind students that logarithmic functions are inverses of exponential functions. As the students slide the ‘a’ slide bar have them take notice of the exponential functions. Point out that these functions are restricted to quadrants one and two; therefore their inverses, the logarithmic functions, must lie within quadrants two and four.

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Now have students set a = 1.

Ask student why the condition a≠ ≠ 1 is in the definition of logarithmic functions. Students should see that the resulting graph is not a function. Algebraic inverses

Have students type e3 into their calculators. Now have them take the ‘ln’ of their answer. Ask students what they notice. Have students make conjectures as to why this happened. To test their conjectures further have them type 107 into the calculator. They should then take the “log" of their answer. Students should see the inverse property at work in the previous two examples. Ask students to evaluate the following expression:

logaaq =____

Using the idea of inverses, students should see the expression simplifies to ‘q’. Back to the millionaire problem

To find out how long it would take to reach $1,000,000, the equation below has to be solved for ‘t’. 1,000,000 = 10,000e0.10t The first step would be to divide both sides by 10,000. 100 = e0.10t Now since e and ln are inverses, we take the ln of both sides. ln(100) = ln(e0.10t)

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Which simplifies to 4.605 = 0.10t 46.05 = t Thus it would take a little over 46 years to accumulate $1,000,000.

Conclusion Logarithmic functions are the inverses of exponential functions. A logarithm is defined as: y = logax if and only if x = ay. The domain of a logarithmic function is limited to positive values of ‘x’. Logarithms are used in a variety of applications, one of which involves compound interest.

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