# Logarithms Lesson 6 (Teachers)

### Justifying the Product Rule for Logarithmic Expressions

Josh and Arobindo write two equivalent exponential expressions that represent the time it takes for the amount of mold to increase by one factor, and then by another. In doing so, they justify a rule for rewriting logarithmic expressions of a certain form.

##### Episode 1: Making Sense

The students model on their number line the mold growth on the pizza over two consecutive time periods. They then calculate the total elapsed time over the two periods of time.

##### Episode 2: Exploring

Josh and Arobindo consider another student’s equation that is meant to describe the mold growth over two consecutive time periods. They decide the student has written an incorrect equation and propose a new equation. They explain why the two logarithmic expressions in their equation are equivalent.

##### Episode 3: Repeating Your Reasoning

Arobindo and Josh fill in the blanks of the equation log3 (___) + log3 (___) = log3 (1,000) to make the equation true. They justify their answer.

##### Episode 4: Reflecting

The students find a logarithmic expression that is equivalent to log3 (x) + log3 (y). They explain what their equation means in terms of the moldy food context and justify their proposed relationship using a number line.

### Mathematics in this Lesson

Targeted Understandings

This lesson can help students:

Use their quantitative understanding of logarithms to justify the product rule for logarithmic expressions (i.e., log3(xy) = log3x + log3y). Specifically, they can interpret the expressions log3x and log3y as the amount of time it takes for the amount of mold to increase by a factor of x and a factor of y, respectively. Similarly, log3(xy) can be interpreted as the amount of time it takes for the amount of mold to increase by a factor of xy, which is the same as the amount of time it takes for the mold to increase by a factor of x and then y.

Common Core Math Standards

CCSS.Math.Content.HSF.BF.5Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.In this lesson, Josh and Arobindo leverage their understanding of logarithms and exponents to create expressions to represent periods of elapsed time in a context about mold that grows on pizza. They move fluently between exponential notation and logarithmic notation as they reason about periods of time composed of several smaller periods of time when they have information about the growth of the mold during those periods of time.

Common Core Math Practices

CCSS.MATH.PRACTICE.MP5Use appropriate tools strategically

According to the CCSSM, “Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations.” Josh and Arobindo make extensive use of their number line in this lesson. The number line, with elapsed time on the bottom and the quantity of mold on the top, seems to act as a powerful tool for reasoning for Arobindo and Josh. For example, in Episode 1, Arobindo inscribes tick marks and arcs on the number line to make an estimate for how long it takes the mold to increase to 5 grams [2:57]. In Episode 3, they use the number line to create annotations that support their argument that log310 + log3100 = log31000.