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# How do we use Logarithms and Exponentials

Lesson 9 of 11

## Objective: SWBAT use formulas involving logarithms to solve problems.

## Big Idea: How can logarithms be used to determine the if the data is exponential or a power function. Applications including decibel, richter scale, pH, Newton's law of cooling,...

*45 minutes*

#### Bell Work

*5 min*

Today students will be reading information about real world uses of exponential and logarithmic functions, including formulas students can use to solve problems. To prepare, I begin this lesson with a question about active reading strategies.

Many of my students unsure how to make a plan when asked to analyze mathematical text and set up a problem. Throughout my course, I try to help students develop techniques for reading text. Today, I ask students to recall what we have covered so far in preparation for their work during the lesson.

Some reading techniques that I make sure the class considers today are:

- re-reading
- highlighting equations
- identify the meaning of the parameters in the formula
- identify confusing words and determine the meaning of the words

When students start working problems, I will encourage them to use these methods:

- identify the formula
- determine the parameter you need to find
- identify the value of the parameters known
- determine the method needed to solve the problem

After students share their ideas and we cover the lists above, we organize the ideas and classify them as understanding the formula and/or using the formula. I encourage students to consider this as a way to organize their notes and review sheets.

#### Resources

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Now that we have discussed some strategies for reading problems with technical content, students receive the first 4 pages of Problem Solving with Exponential and Logarithmic Functions worksheet. I ask my students to work in pairs to read the sections and complete the **Now you try** problems. Students may use their book or a phone if they need to look up a word. If they are still confused by a term after using these resources, they can ask me for clarification. I really want students to rely on each other, and on available information resources, as they work through the activity.

As students work I am making sure students stay on task. I ask clarifying questions such as:

- What is the equation for this type of problem?
- What do the parameters (letters) mean in the equation?
- What are you trying to find?
- What do you know?
- How can find the answer?

If students cannot answer my question, I help them refer back to an appropriate place in the reading. I ask the student to read the information and then will say something like "oh, t represents time. How is that used in solving the problem? What strategies did we talk about for documenting the information in a problem?"

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With about 10 minutes left in class I will ask the students to tell me which topic is the most confusing.

I expect that my students will find Problem 3 on the worksheet difficult. Oftentimes, students do not understand how to write **r **as a function of **t**. When we go over this, I like to begin by reviewing the syntax of function notation and then dive into the development of a formula for r that depends on the time (t) it takes for the mass to be half the original. If possible, I will have one or more students share how to solve the problem.

Another challenging aspect of today's work is understanding that when an initial mass is 1 gram, the remaining amount will be 1/2 gram after decay. The difficulty is when the problem says half the initial amount. Some students find the term initial amount confusing. We define this term and I then say something like,"if I have an initial amount of 10 grams, what is half the initial amount?"

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#### Closure

*5 min*

As class ends, I give students the last 2 pages of Problem Solving with Exponential and Logarithmic Functions as homework. I also let students know that the upcoming assessment will not require them to memorize any of the formulas on the worksheet. I will give students information about a formula and they will need to interpret the information to solve a problem.

As students leave the class, I ask for an exit slip. I ask students to tell me:

- What was most difficult with today's activity?
- what was easiest?

I say, "You must answer both prompts of the question." I know this is an open question and I may get many responses. Asking this question gives students an opportunity to express their frustrations. Many students focus on the negative, by requiring both parts of the questions be answered makes students really think about what they did. Of course some students will not take the question seriously but others will realize they did know how to do something. I think that both outcomes are good for students, but of course the specific details of what is easy and hard is most helpful for planning tomorrow's review.

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- UNIT 1: Introduction to Learning Mathematics
- UNIT 2: Functions and Piecewise Functions
- UNIT 3: Exponential and Logarithmic functions
- UNIT 4: Matrices
- UNIT 5: Conics
- UNIT 6: Solving Problems Involving Triangles
- UNIT 7: Trigonometry as a Real-Valued Functions
- UNIT 8: Graphing Trigonometric Functions
- UNIT 9: Trigonometric Identities
- UNIT 10: Solving Equations
- UNIT 11: Vectors and Complex Numbers
- UNIT 12: Parametric and Polar graphs and equations

- LESSON 1: The graphs of logarithmic and exponential functions
- LESSON 2: Special exponential "e"
- LESSON 3: Evaluating exponential and logarithms
- LESSON 4: Expanding and Condensing logarithms
- LESSON 5: Solving Exponential and Logarithmic Equations
- LESSON 6: Find intercept by solving equations
- LESSON 7: Change of Base Formula
- LESSON 8: Survey and Review Day
- LESSON 9: How do we use Logarithms and Exponentials
- LESSON 10: Review exponential and logarithmic functions
- LESSON 11: Exponential and Logarithm Assessment