# Logarithms Lesson 3 (Teachers)

### Introducing Logarithms

The students explore the definition of a logarithm and describe a logarithm as an exponent in an exponential expression. They interpret logarithms in a moldy pizza context as the time at which a slice has a certain amount of mold.

##### Episode 1: Making Sense

Josh and Arobindo create a number line to model the growth of mold on a pizza slice. They estimate when the pizza will have 50 grams of mold, and mark this in on their number line.

##### Episode 2: Exploring

The students use their number line and a calculator to determine when the pizza slice had 50 grams of mold with more precision.

##### Episode 3: Repeating Your Reasoning

Arobindo and Josh use their number line and a calculator to determine when the pizza slice had 110 grams of mold.

##### Episode 4: Reflecting

Josh and Arobindo find out what a logarithm is. They describe how to use logarithms to find the time when the pizza had any given number of grams of mold.

### Mathematics in this Lesson

Targeted Understandings

This lesson can help students:

• See a logarithm as the exponent you raise a base to, in order to find a particular number (i.e., logba is the exponent you raise b to in order to get a).
• Develop a quantitative understanding of logarithmic expressions in the moldy food context. In particular, students can interpret the logarithmic expressions as the time at which a food has a particular amount of mold (e.g., log350 is the hour at which a food has 50 grams of mold, if it had 1 gram of mold at time 0 and the mold triples each hour).

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.

This lesson features Arobindo and Josh working together to find times at which there are specific amounts of mold on a piece of pizza. They know the mold triples every hour, so they are able to model the situation with the equation 3x = y where y is the amount of mold in grams on the pizza after x hours. However, unlike previous lessons, they do not initially have a way to solve for x. This prompts a guess-and-check method in which they find close approximations for the exponent needed. By the end of the episode, they learn that logarithmic expressions can be used to represent the exact values they had been searching for earlier in the lesson.
• 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.

At the end of the lesson, Josh and Arobindo are introduced to logarithmic notation. They utilize this notation to describe the exact moment there is 65 grams of mold on a piece of pizza. They connect logarithmic expressions to exponential expressions, for example recognizing log365 as the exponent needed to make 3= 65 true.

Common Core Math Practices

CCSS.MATH.PRACTICE.MP6Attend to precision.

According to the CCSSM, “Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately…. They are careful about specifying units of measure…. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context.” This lesson features several examples of Josh and Arobindo attending to precision. For example, in Episode 2 [2:38], when asked to find a more precise time when the pizza had 50 grams of mold, the students use a strategic guess and check to find the exponent x that makes 3= 50 true. Their process allowed them to home in on the value 3.5615, which is close to the value of log350. The students also attend to precision by frequently referring to the units of measure for the quantities they work with. In Episode 2, Arobindo is careful to describe 50 as 50 grams of mold and 3 as the growth factor over one hour [1:54]. They also label the components in the equation 34.279 = 110 using quantities from the context [Episode 3, 1:57].