## Student 7.jpg - Section 2: What is a natural logarithm?

# Puzzling Log Equations

Lesson 5 of 11

## Objective: SWBAT solve basic logarithmic equations by applying log properties.

## Big Idea: Students complete a puzzle in teams to build their skills of applying logarithm properties.

*50 minutes*

As a warm-up today, students will practice condensing a logarithm expression by applying the properties they learned yesterday. There are three clicker questions on pages 2-4 of the Flipchart.

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Before talking about natural logarithms with students, I wanted to pass back their explanations of what* e *is that students wrote in the exponential unit (Day 3 of Credit Card Investigation). Here are a few samples of students responses: Student 4; Student 6; Student 7; Student 10; Student 11;. I want students to review what they have already learned about *e *written in their own words*. *Mainly, I want students to recall that *e *is a mathematical constant (not a variable!) and it arises in situations where continous growth is occuring. I may have students share what they wrote with others at their table so all students can have a variety of explanations of how they interpreted the meaning of *e *in that credit card activity. Then I will present page 5 of the flipchart and have students copy the definition into their Personal Dictionaries.

As students copy the definition, I will inform them that since log base e arises in many situations in mathematics it make sense that we call it a natural logarithm. It’s a bit more special mathematically than the plain logarithm with an implied base 10 as it is universal across all mathematics and in all different numbering systems. Our numbering system is a base of 10 (probably because we have 10 fingers?). But other cultures have based their counting systems on bases such as 5, 8, 12, 20, and 60. So log base 10 would not at all be a ‘natural log’ for those cultures. Natural logarithms are even so special it gets its own abbreviation too, *ln*.

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#### Logarithm Puzzle

*37 min*

**Preparation: **For today’s puzzle students will need scissors to cut apart the pieces and glue/tape to put completed puzzles back together.

**Narrative: **Students will now complete the Logarithm Puzzle. When the puzzle has been solved it should form a triangle (see Puzzle Answer Key). I have found that if students are getting a slow start to the puzzle, it helps to give them the general shape. You may also want to inform them that some of the results don’t match an expression and will end up on the edges of the puzzle. I will have my students glue their completed puzzles on another sheet of paper and submit them for grading. The puzzles are really easy to grade with just a quick look.

I would like students to work in teams to complete the puzzle although each individual student should submit a completed puzzle. I think that by allowing students to work in teams we are going to give students an opportunity to possibly improve on **MP3: Construct viable arguments and critique the reasoning of others.** Also, to help students improve on this practice I will highlight conversations that I hear that I like.

As the students work, I will visit teams and encourage them to explain their thinking to their teammates and justify their solutions using the properties we are studying about logarithms. Also, I do not plan on giving much assistance in regards to the actual solving of the problem. I am going to focus my time on improving communication in the teams, hopefully! So students will definitely be reminded throughout this activity how important it is in mathematics to **MP1: Make sense of problems and persevere in solving them.**

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- UNIT 1: Basic Functions and Equations
- UNIT 2: Polynomial Functions and Equations
- UNIT 3: Rational Functions and Equations
- UNIT 4: Exponential Functions and Equations
- UNIT 5: Logarithmic Functions and Equations
- UNIT 6: Conic Sections
- UNIT 7: Rotations and Cyclical Functions
- UNIT 8: Cyclical Patterns and Periodic Functions
- UNIT 9: Trigonometric Equations
- UNIT 10: Matrices
- UNIT 11: Review
- UNIT 12: Fundamentals of Trigonometry

- LESSON 1: What Is A Log?
- LESSON 2: Graphing and Shifting Logs
- LESSON 3: Discovering Log Rules (because logs rule!)
- LESSON 4: Applying Log Rules
- LESSON 5: Puzzling Log Equations
- LESSON 6: Solving Exponential Equations Using Logs
- LESSON 7: Experts on Exponential Equations
- LESSON 8: Modeling Exponentials Using Logarithms
- LESSON 9: Speed Dating with Logarithms
- LESSON 10: Logarithmic Equations Test Review
- LESSON 11: Logarithmic Equations Unit Test