Unit 4, Lesson 17
Logarithm Power Rule
Integrated Math 3
17.1
Warm-up
Standards Alignment
Building On
Addressing
A-SSE.3
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
F-LE.4
For 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.
Building Toward
Instructional Routines
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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
NoneFor 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
Instructional Routines
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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 , and in this problem? Does your answer from the Warm-up match your conjecture?"
17.3
Activity
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
NoneLet's work through some steps of a proof for your conjecture.
Start with the equation:
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Rewrite the equation as a logarithm, and circle your answer to use later.
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Raise each side of the original equation to the power of .
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Combine the exponents on the left side of the equation so that the left side is written with a single base.
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Rewrite the last equation as a logarithm with a base of .
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Use your circled equation to replace any in that equation with an equivalent logarithm.
Lesson Synthesis
Create a class reference chart for students to refer to throughout this section. Add these logarithm rules to the reference chart.
product rule
quotient rule
power rule
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 . The quotient rule is related to . The power rule is related to .)
- "If you change the role of and in each rule, is it still the same value? For example, is ?" (If , then this works only for the product rule. It works for the product rule because order does not matter for adding or multiplying. In the quotient rule, it does not work because order matters for subtraction and division. In the power rule, it does not work because and it matters which value is inside the logarithm.)
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.