# Logarithms Lesson 5 (Teachers)

### Comparing Interpretations of Logarithms

The students continue to explore the two interpretations of logarithms in the moldy pizza context that they had found before, one as the hour at which the slice has a certain amount of mold and the other as the amount of time it takes for the mold to increase by a certain factor. They interpret several logarithmic expressions in both ways.

##### Episode 1: Making Sense

The students use the definition of a logarithm to say what log3 (10) means generally. They find its value and discuss why that value makes sense, given the definition.

##### Episode 2: Exploring

Arobindo and Josh interpret the expression log3 (10) as the time at which the pizza slice had a certain amount of mold.

##### Episode 3: Exploring

Josh and Arobindo interpret the expression log3 (10) as the time it takes the mold on the pizza slice to increase by a certain factor.

##### Episode 4: Repeating Your Reasoning

The students explore a different moldy food–a peanut butter sandwich. The mold on the sandwich increases by a factor of 4 each hour. The students write a logarithmic expression that describes when the sandwich will have 200 grams of mold.

##### Episode 5: Repeating Your Reasoning

Josh and Arobindo write a logarithmic expression that describes that amount of time it takes the mold on the peanut butter sandwich to increase by a factor of 20.

### Mathematics in this Lesson

Targeted Understandings

This lesson can help students:

• Distinguish between the two quantitative interpretations of logarithms (from Lessons 3 and 4) and explain how these interpretations relate to the definition of a logarithm as the exponent you raise a base to get a particular number.
• Use the two quantitative interpretations of logarithms to model situations where a mold increases by a factor of 4 each hour instead of a factor of 3.

Common Core Math Standards

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.

Josh and Arobindo make sense of a new context, one in which the amount of mold is increasing at a rate of 4 times per hour. The students use a number line to create equations that model mathematical problems in this context. For example, to figure out how long it takes the mold to increase by a factor of 20, they create the equation log420 = x and then make connections between that equation and the equation 4= 20.

Common Core Math Practices

CCSS.MATH.PRACTICE.MP7. Look for and make use of structure.

According to the CCSSM, “Mathematically proficient students look closely to discern a pattern or structure… They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects.” In this lesson, Arobindo and Josh utilize the structure of number lines that represent both the elapsed time that a mold has been growing and the amount of mold that is present. They make connections between the expressions that they write (e.g., log420) and the number lines that they draw. When they create an expression that does not accurately model the time for which it takes the mold to increase by a factor of 20, they utilize the structure of their number line to check their work, realize their error, and make appropriate corrections. They are also able to reason about the complex expression log420 in multiple ways: as an exponent that 4 can be raised to get 20 and also as the length of time needed for mold to increase by a certain factor.