# Logarithms Lesson 4 (Teachers)

### Interpreting a Logarithm as an Elapsed Time

The students figure out that there is another interpretation for logarithms in the moldy pizza context. They describe logarithms as the number of hours it takes for the mold to increase by a certain factor.

##### Episode 1: Making Sense

Josh and Arobindo continue to investigate the growth of mold on a pizza. They use a number line to estimate how long it will take for the mold to increase by a factor of 10.

##### Episode 2: Exploring

The students use their number line and a calculator to determine the amount of time it takes for the mold on the pizza to increase by a factor of 10.

##### Episode 3: Repeating Your Reasoning

The students use their number line and a calculator to determine the amount of time it takes for the mold on the pizza to increase by a factor of 1.5.

##### Episode 4: Reflecting

The students use their number line and a calculator to determine the amount of time it takes for the mold on the pizza to increase by a factor of 1,000,000.

### Mathematics in this Lesson

Targeted Understandings

This lesson can help students:

• Use their understanding of a logarithm as an exponent to develop another quantitative interpretation of a logarithmic expression in the moldy food context. In particular, students can interpret the logarithmic expression as the amount of time it takes for the mold to increase by a certain factor (e.g., log310 is the number of hours it takes for the mold to increase by a factor of 10).

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, Arobindo and Josh utilize logarithmic notation to express the time it takes for mold on a pizza to increase by a given growth factor. For example, they write log31,000,000 to represent the time it takes for mold that increases by a factor of 3 each hour to increase by a factor of 1 million. After finding that logarithm, they translate that expression into exponential notation:  312.57 = 1,000,000.
• CCSS.Math.Content.HSF.LE.A.4For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Throughout this lesson, Josh and Arobindo utilize technology to evaluate technology. Early in the lesson, they use a strategic guess-and-check method to find the length of time it takes the mold to grow by a factor of 10. They use a calculator and the expression 3x to find that x is approximately 2.1. Later they use a logarithmic calculator to find the length of time it takes the mold to increase by factors of 1.5 and 1,000,000.

Common Core Math Practices

CCSS.MATH.PRACTICE.MP2Reason abstractly and quantitatively.

According to the CCSSM, “Mathematically proficient students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved.” This lesson features Arobindo and Josh continuing to explore the moldy pizza context. In doing so they reason both abstractly and quantitatively to build understanding of the mathematical concept of logarithms. For example, in Episode 1 they utilize the context of the moldy pizza to reason quantitatively about the time it takes for the mold to increase by a factor of 10. They argue that since 10 is just a bit bigger than 9, it must be a bit higher than 2 hours, the time for which it takes the mold to increase by a factor of 9 [0:40]. Later, they reason more abstractly by using a series of calculations to increase their precision with their guess for the time it takes [Episode 2].