Unit 4, Lesson 18
Logarithm Change of Base Rule
Integrated Math 3
18.1
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
NoneMaterials
NoneActivity Narrative
In this Warm-up, students use their understanding of logarithms to recognize an identity associated with logarithms and exponents. They must make use of the structure (MP7) of logarithms to understand how to interpret and explain their solutions.
Launch
NoneActivity
NoneUse your understanding of logarithms to find the value of each expression. Be prepared to explain your reasoning.
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of the discussion is for students to recognize a mathematical identity using logarithms and exponent. Because an expression like means "the power to which can be raised to get ," this means that raising to that power equals .
Invite students to share their solutions and reasoning. Ask students:
- "Did you notice a pattern with the solutions? Make a conjecture based on the pattern you noticed." (When the base of the exponent and the base of the logarithm match, the exponent and the logarithm seem to undo each other. It looks like .)
- "Does the pattern work in the other order? That is, is it true that ?" (Yes, using the power rule, the left side of the equation is equivalent to , and .)
Display both identities for all to see, and leave them displayed for the rest of the lesson.
Tell students that these equations are always true, but do not need to be proven because it comes directly from the definition of a logarithm.
18.2
Activity
Instructional Routines
NoneMaterials
To Gather
Scientific calculatorsActivity Narrative
In this activity, students play with using the logarithm rule for changing the base of the logarithm. This technique is useful for calculating logarithms that have a base other than 10 or and are not integers.
Launch
Arrange students in groups of 2. Provide access to scientific calculators.
Display the change of base rule for logarithms for all to see.
Tell students that this is called the change of base rule for logarithms, then ask students why they think it is called that. (because the base of the logarithm changes)
Tell students that the new base, , can be any positive number other than 1. If students ask why 1 is not allowed, direct them to the Are You Ready for More? questions.
18.3
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this activity, students complete the steps of a proof for the change of base rule for logarithms.
Launch
Arrange students in groups of 2.
Ask students which logarithm rules they have already proven. If necessary, prompt them to state the power rule: .
Display the change of base rule, , and tell students they will work through a proof of this rule.
Lesson Synthesis
The purpose of the discussion is for students to understand the structure of the change of base rule and its common use as a way to approximate logarithm values.
Ask students,
- “What happens if you use the change of base rule to change into logarithms with base ?” (It is equivalent to . Then, because , this is the same as .)
- “One of the most common uses of the change of base rule is to approximate values for logarithms in any base. If your calculator can find values for logarithms only in base 10 or base , how can you approximate the value of an expression like ?” (I can change the base to either base 10 or base to get and then find that this is approximately 6.1699.)
- “Many computer algorithms that put a large group of numbers in order work by first dividing the group in half, then doing something to each half. This means that computer scientists sometimes use logarithms in base 2 to look at how fast their algorithms work. If you write a different algorithm that divides the group into thirds, how could you change a base-2 logarithm into a base-3 logarithm?” (Using the change of base rule, .)
Student Lesson Summary
The change of base rule allows us to rewrite any logarithm to have any other positive value (except 1) for the base. We can use this equation to change the base: .
Many calculators can calculate logarithms only in base 10 () or base (). The change of base rule can be used to change any logarithm into one of these bases so we can find approximate values of any logarithm.
For example, we know that is between 2 and 3 because and , but it requires a lot of guessing and checking to get closer to an actual value when our calculator can't compute logarithms in base 2. Using the change of base rule allows us to rewrite the logarithm as , and this can be entered into most scientific calculators to get about 2.3219. We could have also changed into logarithms with base to get , which has the same approximate value.