Unit 4, Lesson 17
Logarithm Power Rule
Integrated Math 3
17.1
Warm-up
10 mins
A Pattern with a Logarithm Product
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
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this Warm-up, students examine products of logarithms with a rational numbers to find a pattern. All of the logarithms in this activity have integer values, so no technology should be necessary to calculate the logarithms, but calculators may help with finding the value of larger powers.
As students identify the pattern, they are expressing regularity in repeated reasoning (MP8).
Launch
Arrange students in groups of 2.
Activity
NoneStudent Task Statement
For each equation, find the value of the missing terms by finding the value of the logarithms and comparing the values on each side of the equation.
The first one is done for you. Discuss with your partner why it is true.
Student Response
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Building on Student Thinking
Activity Synthesis
Invite students to share the values and logarithms they wrote for each problem. Then ask students,
- “What do you notice about the logarithm of an expression with an exponent and the product you wrote?” (The base of the logarithm and the base of the logarithm in the product are unchanged. The exponent of the expression inside the logarithm is the same as the value multiplied by the logarithm in the product.)
- "Is there anywhere else you have seen products and powers related in a rule before?" (There is an exponential rule that says
17.2
Activity
10 mins
Making a Conjecture about a Logarithm Product
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this activity, students make a conjecture based on the pattern they noticed in the Warm-up. They use that conjecture to rewrite and calculate logarithms that would not be possible for them to do otherwise.
Launch
Arrange students in groups of 2.
Give students 2–3 minutes to write their conjectures, then pause the class to discuss the patterns students noticed. Invite groups to share their conjectures and compare the conjectures to the problems from the Warm-up. Select a problem from the Warm-up and ask, "What are the values of
17.3
Activity
15 mins
Proving the Conjecture about a Logarithm Product
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this activity, students prove their conjecture about the product of a logarithm with a rational number. In the whole-class discussion, students are told that this pattern is called the power rule for logarithms.
Launch
NoneActivity
NoneStudent Task Statement
Lesson Synthesis
Create a class reference chart for students to refer to throughout this section. Add these logarithm rules to the reference chart.
The purpose of the discussion is to understand the connection to exponent rules and highlight important aspects of the ordering of variables in the rules.
Ask students,
- “How could we restate each rule in words?” (Product rule: The sum of two logarithms with the same base is equal to the log of the product of the arguments. Quotient rule: The difference of two logarithms with the same base is equal to the log of the quotient of the arguments. Power rule: The logarithm with an argument raised to a power is equal to the product of the power and the logarithm with its argument raised to the power of 1.)
- "What exponent rule is related to each of the rules?" (The product rule is related to
- "If you change the role of
and in each rule, is it still the same value? For example, is
If time permits, additional connections among the logarithm rules can be presented. Display these equivalent expressions one pair at a time, then ask students to explain why they are equivalent.
Student Lesson Summary
The power rule for logarithms allows us to rewrite logarithms with values raised to powers. The power rule states that
For example,
Thinking about logarithms in relation to exponents, this may make more sense. We learned in an earlier course that
By rewriting parts of that equation into their logarithm form, we can combine the pieces to prove the power rule.