Unit 4, Lesson 11
Evaluating Logarithmic Expressions
Integrated Math 3
11.1
Warm-up
Standards Alignment
Building On
Addressing
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.
Materials
NoneActivity Narrative
This Math Talk focuses on evaluating logarithms. It encourages students to think about the meaning of logarithms and to rely on the structure of the notation to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students solve equations that include a logarithm.
To evaluate the logarithms, students need to look for and make use of structure (MP7).
Launch
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
- Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
- Invite students to share their strategies, and record and display their responses for all to see.
- Before moving to the next problem, use the questions in the Activity Synthesis to involve more students in the conversation.
Keep all previous problems and work displayed throughout the talk.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization
Supports accessibility for: Memory, Organization
Activity
NoneEvaluate mentally.
Activity Synthesis
Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:
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“Who can restate ’s reasoning in a different way?”
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“Did anyone have the same strategy but would explain it differently?”
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“Did anyone solve the problem in a different way?”
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“Does anyone want to add on to ’s strategy?”
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“Do you agree or disagree? Why?”
If not already mentioned, make sure students recall that numbers such as 0.1 and are still powers of 10 but they have negative exponents (that is, 0.1 is , and is or ). This suggests that the logarithms of numbers that are less than 1 would be negative.
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example:
Advances: Speaking, Representing
- "First, I _____ because _____.”
- "I noticed _____, so I _____.”
Advances: Speaking, Representing
11.2
Activity
Instructional Routines
MLR8: Discussion SupportsMaterials
To Copy (from Blackline Masters)
Log War Cards
Activity Narrative
In this activity, students compare logarithmic expressions as they play a game of “war” in groups of 2. Students receive cards containing logarithms in bases 2, 5, and 10. The numbers in the expressions are written in all forms, including decimals, fractions, whole numbers, and exponential expressions. To compare the logarithms on the cards, they may:
- Determine if an expression is negative, zero, or positive, without finding the exact or estimated value.
- Estimate the value of an expression such as by using logarithms with integer values.
- Convert decimals to fractions to help find the logarithms.
As students preview the deck by putting the cards in order from least to greatest, listen for how they decide on the relative values of the cards. Students should not use calculators at this point.
11.3
Activity
Instructional Routines
NoneMaterials
To Gather
Scientific calculatorsActivity Narrative
This activity has two aims: to familiarize students with the log function on a calculator, and to reinforce the fact that most values obtained on a calculator, like those in a log table, are approximations.
The questions involve finding only base-10 logarithms. This is partly to focus the exploration, and partly because many calculators can evaluate only logarithms in base 10 and base , or require the user to perform steps to change the base before calculating logarithms in other bases. Some calculators allow easily calculating logs with any base. If the devices used in your class allow for this, show students how it works.
The purpose of the last question is to prompt students to think about reasonable values of the logarithms and to not simply enter numbers into the calculator. Monitor for students with clear estimation strategies. Beyond reasoning that a log value is between two whole numbers, students don't yet have a good means of making an accurate estimate, which is fine. For example, they might be able to say that should be quite close to 2, but it is not expected that they could say how close.
Lesson Synthesis
Encourage students to reflect on their strategies for finding or estimating logarithms. Discuss questions such as:
- “When playing the game, how did you know if the value of the expression is positive, negative, or zero?” (If the logarithm is of the form and , then the logarithm is positive. If , then the logarithm is zero. If , then the logarithm is negative.)
- “When the solution to an exponential equation is estimated with a calculator or a log table and not written in log notation, how would you know if it’s reasonable? For example, is 2.675 a reasonable value for ?” (No, the correct value should be a little greater than 4. The value of is the exponent for the base of 2 that results in 18. I know that and , so the exponent for the base of 2 that results in 18 should be between 4 and 5, closer to 4.)
Student Lesson Summary
Sometimes it’s possible to find an exact value for a logarithm. For example because , and . Similarly, because .
Often it is not possible to find an exact value, but using number sense allows us to get a reasonable estimate. Let’s say we want to find that makes true.
We can first express as . Then, because 980 is between and we know that must be between 2 and 3.
But where does lie between 2 and 3? Because 980 is much closer to 1,000 than it is to 100, is likely a lot closer to 3 than it is to 2. This means that 2.9 would be a better estimate than 2.1 would be.
We can use a calculator to verify our estimate and find that is about 2.99.