Unit 4, Lesson 18
Logarithm Change of Base Rule
Integrated Math 3
18.1
Warm-up
10 mins
Log as a Power
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 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
NoneStudent Task Statement
Use 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
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
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
10 mins
Changing the Logarithm Base
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
18.3
Activity
15 mins
Proving the Change of Base Rule
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,
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 .) - “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 - “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 (
For example, we know that