##
* *Reflection: Connection to Prior Knowledge
Getting to Know Logarithms - Section 2: Getting to Know Logarithms

The key to success with logarithms is a solid conceptual understanding. Students are more successful and find mathematics more enjoyable when they can see the subject as a coherent and connected whole. This doesn't mean that they've be *told* that logarithms are related to exponents, but that they've had the opportunity to really *see* the connection between the two concepts.

For this view of mathematics to form, students must construct it for themselves out of many different experiences. Gradually, as new concepts are observed to develop out of prior concepts, the cumulative effect is a clear and coherent view of the subject.

When it comes to logarithms, we begin with exponentiation. From this foundation, we slowly develop the language of logarithms, the notion of a logarithm as the inverse of an exponent, and the various properties of logarithms. Along the way, we continually reinforce the connections between the concepts and pick up lots of practice.

Since I moved toward this approach, I've had lots of success, with students even coming back to say that logarithms were their favorite subject in Algebra 2 because they made so much sense!

*The Key to Success*

*Connection to Prior Knowledge: The Key to Success*

# Getting to Know Logarithms

Lesson 10 of 14

## Objective: SWBAT rewrite exponential expressions in logarithmic form. SWBAT apply the properties of logarithms.

#### What is a Logarithm?

*10 min*

*Be advised: This lesson begins to address (+) standards for Algebra 2 because I ask students to use logarithms to solve exponential equations with bases other than 10, e, or 2.*

As briefly as possible, we will recap the previous lesson. I'll ask some pointed questions to help the students recall the following points:

1. We *postulate* that every number is a power of every other number. This postulate is what makes the logarithms possible.

2. "Taking a logarithm" is the inverse of exponentiation. It's the operation we use when we're trying to find an unknown exponent. The *logarithm* is the exponent that we're looking for. (Say, how did that homework go? Did you solve 3^x = 17 and 3^x = 23?)

3. We can use logarithms to turn division into subtraction ("Would someone please show me how it's done with 23 divided by 17.") and multiplication into addition ("Did someone figure this out last night? Please show us.").

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#### Getting to Know Logarithms

*25 min*

Today's main goal is to get comfortable with logarithms. Practice will help this to happen, as well as continual reference back to familiar exponents.

Now, I hand out Getting to Know Logs and ask the students to begin working independently. For the next 10 minutes or so I'll move around the room to check for understanding. The first problems are intended to get students used to the way we speak about logarithms, as well as the interpretation of the symbols we use. (**MP 6, see video**)

Once I can see that everyone understands how solve these problems and they're about 2/3 of the way through the front, I'll announce that they may now begin collaborating if they want. Their first task, as always, should be to compare answer and resolve any differences. (It might be a good idea here to quick review the answers to the first 10 or 15 problems. I'd ask one student after another to read the complete sentences and ask the rest of the class if they agree.)

In the time that students have to work today, I expect that many of them will complete the front of the handout and make it through part of the back. When they begin working on the back, watch what they're doing with their calculators to make sure that they're working within the given constraints. Otherwise, they'll get no benefit from this problem set!

Check out this annotated version for the rationale behind the different problems.

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#### Logarithm Sprints

*10 min*

Before beginning these first Logarithm Sprints, you should make absolutely certain that everyone understands how to "translate" and exponential equation into a logarithmic one and vice versa. Choose one or two examples to show on the board, such as 2^3 = 8 or 3^4 = 81. Also, show at least one example of a common logarithm, such as 10^(-2) = 0.01.

Finally, give an example problem like this: and show that it is equivalent to the familiar problem 2^x = 16. In other words, the problems on these sprints only *look* new. (**MP 6: "make explicit use of definitions"**)

Now, end class with logarithm sprint 1 and logarithm sprint 2. Some students may be disappointed with the results, but reassure them that the new expressions require them to think differently, which is why they're moving more slowly. With a little practice, they'll improve greatly!

I typically assign any unfinished problems from the sprints for HW. I'll also set a baseline for completion of the "Getting to Logs" worksheet. For example, I might require that everyone finish the front and half of the back before the next class.

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- UNIT 1: Modeling with Algebra
- UNIT 2: The Complex Number System
- UNIT 3: Cubic Functions
- UNIT 4: Higher-Degree Polynomials
- UNIT 5: Quarter 1 Review & Exam
- UNIT 6: Exponents & Logarithms
- UNIT 7: Rational Functions
- UNIT 8: Radical Functions - It's a sideways Parabola!
- UNIT 9: Trigonometric Functions
- UNIT 10: End of the Year

- LESSON 1: Opening Day Exponential and Logarithmic Functions Unit
- LESSON 2: Rational Exponents
- LESSON 3: Simplifying Exponential Expressions, Day 1
- LESSON 4: Simplifying Exponential Expressions, Day 2
- LESSON 5: Graphing & Modeling with Exponents
- LESSON 6: Comparing Growth Models, Day 1
- LESSON 7: Comparing Growth Models, Day 2
- LESSON 8: Comparing Growth Models, Day 3
- LESSON 9: Logarithms - Napier's Wonderful Invention!
- LESSON 10: Getting to Know Logarithms
- LESSON 11: Properties of Logarithms Day 1 of 3
- LESSON 12: Properties of Logarithms, Day 2 of 3
- LESSON 13: Properties of Logarithms, Day 3 of 3
- LESSON 14: The Number e