Exponential Modeling and Logarithms

Lesson 12

Math

Unit 5

11th Grade

Lesson 12 of 16

Objective


Develop and use the product and quotient properties of logarithms to write equivalent expressions.

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.

Criteria for Success


  1. Apply the product and quotient properties to rewrite logarithmic expressions.
  2. Explain why the product and quotient properties hold.
  3. Connect the product and quotient properties to graphical transformations of logarithmic functions.
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Anchor Problems


Problem 1

Allison says that:

$${\mathrm{log}1000}$$

$${=\mathrm{log}\left(10\cdot{100}\right)}$$

$${=\mathrm{log}10+\mathrm{log}100}$$

Show how Allison's reasoning is correct.

Guiding Questions

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Problem 2

Allison says that:

$${log 100}$$

$${=log\left(1000\over{10}\right)}$$

$${=log1000-log10}$$

Show how Allison's reasoning is correct.

Guiding Questions

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Problem 3

Below are two graphs of logarithmic functions.

$${f(x)=\mathrm{log}_2x}$$

$${g(x)=\mathrm{log}_2(4x)}$$

What transformation occured? How can you use the rule you developed in the last problem to explain the transformation?

Guiding Questions

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Target Task


  1. Write an expression that is equivalent to the expression shown but as the sum of two separate logarithmic terms.

$${{\mathrm{log}_312}}$$

  1. Write a second expression that is equivalent to $${{\mathrm{log}_312}}$$ but as the different of two separate logarithmic terms.
  2. Write a single logarithm that expresses a shift vertically down by $$3$$ units in a single term. Then, show how you would write this log as two terms.

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 problems that ask students to make comparisons of logarithms, similar to the activity Logarithm Clothesline in this blog post.
  • Include problems using both product and quotient rules, as in, "Rewrite as a single log: $${\mathrm{log}25+\mathrm{log}2-\mathrm{log}5}$$."
  • Include problems ientifying equivalent variable expressions, such as $${\mathrm{log}x+\mathrm{log}y=\mathrm{log}(xy)}$$.
  • Open Middle Logs 2Use the first two parts of this problem
  • Rich Starting Points for A Level Mathematics Risp 31: Building Log EquationsModel problems off of this activity, asking whether certain equations are always true, sometimes true, or never true
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Lesson 11

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Lesson 13

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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