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# Logarithms

Lesson 3 of 11

## Objective: SWBAT evaluate logarithms and understand the inverse relationship between logarithms and exponential expressions.

## Big Idea: Logarithms are exponents! To evaluate a logarithmic expression, we ask "what exponent do we put on this number to get that number?"

*85 minutes*

#### Warm-up

*15 min*

My students always need lots of practice rewriting expressions with the properties of exponents, so for today's warm up I ask them to complete Row Game Exponent Rules.docx. Students work in pairs to complete a worksheet with two columns: A and B. Question 1A and 1B are different, but they have the same answer. This is the same for 2A and 2B, etc. As students work through their own column, they stop to check answers with their partner.

#### Resources

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Logarithms are not included in the algebra 1 curriculum at my school, so typically students do not have any familiarity with them when they come to my class. I do not consider this a topic that students will be able to discover for themselves, so I am less likely to approach in through an inquiry activity.

For that reason, I approach the topic of logarithms through a discussion/lecture approach. I write "inverse operations" on the board and students for examples of inverse operations, along with an example. Some of the things that come up might be

- adding 2 and subtracting 2; a+2 = b means b - 2 = a
- multiplying by 2 and dividing by 2, 2*a=b means b/2 = a
- squaring and taking the square root, a^2 = b means sqrt(b) = a, (
*as long as a is not negative)*

Then I will offer

- raising two to this power and ???

in order to lead the discussion to logarithms. I tell them that the inverse of exponentiation is taking the logarithm. We run through a few numerical examples that both reinforce the new concept and help students practice the properties of exponents.

When my students seem to understand what a logarithm is, I show them that any exponential expression can be rewritten as a logarithm.

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#### Active Practice

*20 min*

Because logarithms are new to my students, I like to set aside time for some active practice with them before moving ahead. Logarithm war is like the traditional card game, except that each card contains a logarithmic expression. Students evaluate each logarithmic expression in order to determine who wins the round.

#### Resources

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Now that students understand what a logarithm is, we return to some of the simple equations from the previous day's homework that my students did not have the skills to solve. The final two problems on the homework could not be solved exactly without logarithms, although many students will have estimated the solutions well.

I write these problems on the board and ask students to share how they approached them. For the problem 2^x=6, students might have said 2.4 or 2.5 and we could check how close their answers were by "plugging them in" and seeing how close the left side was to 6. I tell them that the way to solve it exactly is to use logarithms [MP6].

We go through a few examples on the board of using logarithms to provide some exact solutions of exponential equations. Although many textbooks teach students to solve in terms of base 10 or base e, modern calculators can provide approximations for logarithms of any base. Because logarithms are a minor topic in the Common Core State Standards, I do not spend much time developing the concept. My goals for Algebra 2 coverage of logarithms are to make sure that students can

- understand that logarithms are exponents and that "taking the log base b" is the inverse of "raising b to the power"
- translate between exponential notation and logarithmic notation F.BF.B.5
- solve an exponential equation using logarithms F.LE.A.4

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Students will complete Solving Exponential Equations with Logarithms.pdf as homework. This is a collection of equations to solve that require the use of logarithms. I instruct my students to provide "exact answers" and then approximate to a decimal. I want my students to understand that in math class we provide exact answers so that the answer can be rounded with the appropriate level of precision when solving an applied problem [MP6].

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- LESSON 1: Properties of Exponents
- LESSON 2: Exponential Equations
- LESSON 3: Logarithms
- LESSON 4: Quiz and Graphing Exponential Functions
- LESSON 5: Multiple Representations of Exponential Functions
- LESSON 6: Quiz and Comparing Linear, Exponential Models
- LESSON 7: Drinking and Driving Activity
- LESSON 8: Exponential Models in the Sciences
- LESSON 9: Exponential Functions in Finance
- LESSON 10: Review Workshop: Exponential Functions and Logarithms
- LESSON 11: Unit Test: Logarithms and Exponential Functions