Exponential Modeling and Logarithms

Lesson 13

Math

Unit 5

11th Grade

Lesson 13 of 16

Objective


Develop and use the power property of logarithms to write equivalent expressions.

Common Core Standards


Core Standards

  • F.BF.B.4.B — Verify by composition that one function is the inverse of another.
  • F.LE.A.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.

Criteria for Success


  1. Apply the power rule to rewrite logarithmic expressions.
  2. Explain why the power rule holds.
  3. Use multiple logarithm properties to recognize equivalent expressions.
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Anchor Problems

25-30 minutes


Problem 1

Each logarithm below is written two ways. Evaluate each logarithm and identify any patterns in the table.

$${{{{\mathrm{log}1}0}0}0}$$ $${\mathrm{log}(10^3)}$$  
$${{{\mathrm{log}1}0}0}$$ $${\mathrm{log}(10^2)}$$  
$${{\mathrm{log}1}0}$$ $${\mathrm{log}(10^1)}$$  
$${\mathrm{log}1}$$ $${\mathrm{log}(10^0)}$$  
$${\mathrm{log}{1\over10}}$$ $${\mathrm{log}(10^{-1})}$$  
$${\mathrm{log}{1\over100}}$$ $${\mathrm{log}(10^{-2})}$$  

Guiding Questions

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References

EngageNY Mathematics Algebra II > Module 3 > Topic B > Lesson 10Opening Exercise

Algebra II > Module 3 > Topic B > Lesson 10 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

Below is a statement:

$${\mathrm{log}_42={1\over2}\mathrm{log}_44}$$

Is this true? How do you know?

Guiding Questions

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Problem 3

Show that $${y=\mathrm{ln}e^x}$$ is equivalent to $${y=x}$$.

Guiding Questions

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Target Task

5-10 minutes


Problem 1

Establish the property $${\mathrm{log}\left(1\over x\right)=-\mathrm{log}(x)}$$ for all $${x>0}$$ using the properties you have learned thus far.

References

EngageNY Mathematics Algebra II > Module 3 > Topic B > Lesson 12Exit Ticket, Question #1

Algebra II > Module 3 > Topic B > Lesson 12 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

Write each logarithm as an equivalent expression involving only logarithms base 10.

a.     $${\mathrm{log}_3(25)}$$

b.     $${\mathrm{log}_{100}(x^2)}$$

References

EngageNY Mathematics Algebra II > Module 3 > Topic B > Lesson 13Exit Ticket, Question #2

Algebra II > Module 3 > Topic B > Lesson 13 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Problem 3

Rewrite each expression as an equivalent expression containing only one logarithm.

a.     $${3\mathrm{ln}(p+q)- 2\mathrm{ln}(q) - 7\mathrm{ln}(p)}$$

b.     $${\mathrm{ln}(xy)-\mathrm{ln}\left({x\over y}\right)}$$

References

EngageNY Mathematics Algebra II > Module 3 > Topic B > Lesson 13Exit Ticket, Question #3

Algebra II > Module 3 > Topic B > Lesson 13 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Additional Practice


The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Do the graphical illustration of the power property and quotient property (just like Anchor Problem #3 in the previous lesson). Ensure that students can describe the transformation of the log function based on these properties. 
  • Include problems where students must use multiple properties to rewrite, such as rewriting $${2\mathrm{log}_26-\mathrm{log}_29}$$ as a single log.

Next

Solve equations with logarithms.

Lesson 14
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Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Modeling with and Interpreting Exponential Functions

Topic B: Definition and Meaning of Logarithms

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