Unit 4, Lesson 15
Logarithm Product Rule
Integrated Math 3
15.1
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
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NoneActivity Narrative
In this Warm-up, students examine sums of logarithms with the same base 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 sum, find the value of the sum by finding the value of each logarithm and then adding the values. Then complete the given logarithm so that it has the same value as the sum.
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 original sum of logarithms and the single equivalent logarithm you wrote?" (The bases are the same. The product of the values inside the parentheses of the logarithms from the sum becomes the part inside the parentheses for the logarithm I wrote.)
- "Is there anywhere else you have seen sums and products related in a rule before?" (There is an exponential rule that says .)
15.2
Activity
Instructional Routines
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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?"
15.3
Activity
Instructional Routines
NoneMaterials
To Gather
Internet-enabled deviceActivity Narrative
In this activity, students prove their conjecture about sums of logarithms with the same base. In the whole-class discussion, students are told that this pattern is called the product rule for logarithms.
Launch
NoneActivity
NoneLesson Synthesis
Create a class reference chart for students to refer to throughout this section. Add the product rule to the reference chart.
Ask students:
- “How could we restate the product rule in words?” (The sum of two logarithms with the same base is equal to the log of the product of the arguments.)
- “Give an example of an expression that becomes possible to evaluate mentally by applying the rule.” (One example is .)
- “How is the product rule for logarithms related to the exponent rule when multiplying expressions with the same base?” (In both rules, a product can become a sum when the bases of the logarithms or exponents are the same.)
- “Can the sum of three (or more) logarithms be written as a single logarithm using this rule? For example, is ?” (Yes, we can take the logarithms 2 at a time. In this example, we could combine the first 2 logarithms to get , then combine those to get .)
Student Lesson Summary
The product rule for logarithms allows us to combine a sum of logarithms with the same base into a single logarithm. The product 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 product rule.