Unit 4, Lesson 19
Using Logarithm Rules
Integrated Math 3
19.1
Warm-up
Standards Alignment
Building On
Addressing
HSA-SSE.B.3
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
HSF-LE.A.4
For exponential models, express as a logarithm the solution to where , , and are numbers and the base is 2, 10, or ; evaluate the logarithm using technology.
Instructional Routines
Which Three Go Together?Materials
NoneActivity Narrative
This Warm-up prompts students to compare four logarithm expressions. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.
Launch
Arrange students in groups of 2–4. Display the expressions for all to see. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three expressions that go together and can explain why. Next, tell students to share their response with their group and then work together to find as many sets of three as they can.
Activity
NoneWhich three go together? Why do they go together?
A
B
C
D
Student Response
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Building on Student Thinking
Activity Synthesis
Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations, and ensure the reasons given are correct.
During the discussion, prompt students to explain the meaning of any logarithm terminology they use, such as "product rule," "quotient rule," and "power rule," and to clarify their reasoning as needed. Consider asking:
- “How do you know_____?”
- “What do you mean by _____?”
- “Can you say that in another way?”
19.2
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this activity, students use logarithm rules to combine multiple logarithms into a single logarithm. Students may first attempt to combine logarithms only two at a time, then recognize patterns to find the combinations quicker.
Launch
Arrange students in groups of 2.
Display these expressions and ask students to write equivalent expressions.
Although students may suggest many expressions, ensure they see the solutions here.
- This is the same expression as in the last question.
Tell students that, in general, most operations like adding, subtracting, multiplying, or dividing combine only 2 values at a time. Computing a value like is really done as something like .
19.3
Activity
Instructional Routines
Card SortMaterials
To Copy (from Blackline Masters)
Card Sort: Logarithm Rules Cards
Activity Narrative
Students sort different expressions during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7).
Launch
Tell students to close their books or devices (or to keep them closed). Arrange students in groups of 2 and distribute pre-cut cards. Allow students to familiarize themselves with the representations on the cards:
- Give students 1 minute to place all the cards face up and start thinking about possible ways to sort the cards into categories.
- Pause the class and select 1–3 students to share the categories they identified.
- Discuss as many different categories as time allows.
Lesson Synthesis
The purpose of the discussion is to ensure students understand all of the logarithm rules and how they can work together.
Ask students to state the four logarithm rules they learned, and display each one as it is described.
Then ask students:
- "When combining logarithms, what did you notice about which numbers ended up in the numerator and which numbers ended up in the denominator of the combined logarithm argument?" (If the logarithm is added, it is part of the numerator, and if it is subtracted, it will go in the denominator.)
- "How could you break apart a logarithm like ?" (Among other ways, it could become .)
Student Lesson Summary
Logarithm rules can be used in combination to write equivalent expressions. For example, the expression is equivalent to by the change of base rule. Then, using the product and quotient rules, this expression is also equivalent to .