# What Is A Log?

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## Objective

SWBAT convert between exponential and log functions, and formally define the log function.

#### Big Idea

A silent board game leads students to the formal definition of logarithm.

## Warm-up: Silent Board Game

15 minutes

Today’s warm-up requires the teacher to lead it… so beginning of class routines will have to wait a bit for me. Using pg. 2 of today’s flipchart, I am going to give students a little bit of history of logs and emphasize that there is some connection between logarithmic functions and exponential functions, but leave it at that. It is very important that students develop their own understanding that a log is an inverse of an exponential, so I will definitely not be stating that fact yet.

Next, we are going to play a little game as a class (maybe make it a competition between classes?) to see if students can develop the function represented by the table on page 3 of the flipchart. The catch? They can’t talk! Students will be asked to come up the board one at a time and fill in any values they think they know. If they are correct I will leave it. If not, I will erase it. Once a student gets an entry correct they are no longer allowed to fill in anymore. Once we have a complete table, the ‘game’ progresses to the class finding the rule in words or symbols. I predict that someone in class will be able to lead us to x equals 5 to the power of y eventually. If students get horribly stuck I may show them a few more values. If participation/motivation is low… try giving one student the whiteboard marker and then having them pass it off. Make them choose who goes up! Students should be improving on their ability to identify the structure and make use of this structure (MP7) when identifying this equation. It is key that students recognize the square root as an exponent of ½. Also, their reasoning skills are definitely going to get exercised here, particularly quantitative reasoning. So this is a great time to reinforce/remind students about MP2: Reason quantitatively and abstractly.

You will probably want to keep the table of value written on the board for students to reference later when they work on the what is a log worksheet.

## Developing Definition of a Logarithm

20 minutes

Next, I will lead the class though a brief discussion about what seems different with this function. I will give students a moment to share out ideas on how we could reverse this. I am hoping to debunk some common misconceptions here. I am thinking some students will want to divide by 5 or take maybe the fifth root of both sides of the equation. Eventually we will come to a standstill (or someone will suggest a log). So we will now establish the definition of a log both formally and informally. Students should add this definition to their Personal Dictionaries.

As we dive into the world of logarithms, it is essential that students connect logarithms as an inverse to an exponential function.  Most of my students will have already had some experience with logarithms in their past Algebra classes, but I find that they still don’t really understand the concept of a logarithm. So throughout this unit, I will be emphasizing over and over again to students that logarithms are just exponents. If they learn nothing else this unit…. I really hope they can retain that “logs are exponents.” But of course they will learn so much more!

## Applying Definition of Logarithm

10 minutes

Students will now work with applying the definition of a log in a whole group setting so they can receive immediate feedback as they first start to apply this definition. Students should answer the questions on pages 9 and 10 of the flipchart. Collect answers using personal response systems and clarify misconceptions on an individual or whole class basis. Then have students complete the half sheet worksheet, What is a Log? I plan to have students text in select answers using a poll to insure completion.

## Closure: What is a log?

5 minutes

To close out today’s learning students are going to answer the question what is a log to demonstrate their knowledge of the learning target. I expect answers like “it’s an exponent” and the definition. I want to lead a quick talk about which would be a more accurate answer. 0f course the phrase “it’s an exponent” would be missing some detail about the base so I think it’s important that student recognize this.

## Homework

Assign worksheet #1 from this unit for homework tonight.