Exponential Modeling and Logarithms

Lesson 14

Math

Unit 5

11th Grade

Lesson 14 of 16

Objective


Solve equations with logarithms.

Common Core Standards


Core Standards

  • 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.

Foundational Standards

  • A.REI.A.1
  • A.SSE.A.2

Criteria for Success


  1. Use the definition of a logarithm to solve equations.
  2. Identify exponentiation and taking the log of both sides as truth-preserving transformations.
  3. Use exponentiation and taking the log of both sides to solve equations.
  4. Identify appropriate strategies to solve equations involving exponents or logarithms.

Tips for Teachers


  • The problem set will have several variants that require students to manipulate the logarithmic or exponential equation to “reveal” structure or processes that can be used to help solve the problem. 
  • Students should have their properties out so they can have a list of possible moves to make in order to solve the problems. 
  • In this lesson, I opted to focus on the skills of solving equations, doing modeling in the next two lessons.
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Anchor Problems


Problem 1

Explain how you know that these two properties are true:

If $${b^x=b^y}$$, then $${{x=y}}$$ when $${b\neq 0}$$ and $${{b\neq 1}}$$.

If $${\mathrm{log}_bx=\mathrm{log}_by}$$, then $${{x=y}}$$ when $${b\neq0}$$ and $${{b\neq 1}}$$.

Solve $${5^x=5^{3x-4}}$$.

Guiding Questions

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Problem 2

Solve.

$${\mathrm{log}_6(x+3)+\mathrm{log}_6(x-2)=1}$$

Guiding Questions

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Problem 3

Take the natural log of both sides and solve.

$${e^{2x-1}=4e}$$

Guiding Questions

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Target Task


Find all solutions to the following equations. Remember to check for extraneous solutions.

a.     $${\mathrm{log}_2(3x+7)=4}$$

b.     $${\mathrm{log}(x-1)+\mathrm{log}(x-4)=1}$$

References

EngageNY Mathematics Algebra II > Module 3 > Topic B > Lesson 14Exit Ticket, Question #1 and #2

Algebra II > Module 3 > Topic B > Lesson 14 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.

  • Include basic logarithmic and exponential equations first, then get more complicated, for instance, $${\mathrm{log}_2(2x-2)=4}$$.
  • Vary exponential and logarithmic problems.
  • Show a logarithmic equation graphically as a system and have students describe how they know their answer is reasonable based on the graph. 
  • Show an exponential equation graphically as a system and have students describe how they know their answer is reasonable based on the graph.
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Lesson 13

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Lesson 15

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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