Unlocking Logarithmic Equations: The One-to-One Property

Algebra 2 Grades High School 1:38 Video
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Lesson Description

Master the one-to-one property of logarithms to solve logarithmic equations. Learn to simplify and solve equations with common and natural logs, and identify extraneous solutions.

Video Resource

One-to-One Property to Solve Logarithmic Equations

Mario's Math Tutoring

Duration: 1:38
Watch on YouTube

Key Concepts

  • One-to-One Property of Logarithms
  • Logarithmic Equations
  • Extraneous Solutions
  • Quotient Property of Logarithms

Learning Objectives

  • Apply the one-to-one property of logarithms to solve logarithmic equations.
  • Simplify logarithmic expressions using properties of logarithms before applying the one-to-one property.
  • Identify and eliminate extraneous solutions in logarithmic equations.

Educator Instructions

  • Introduction (5 mins)
    Briefly review the definition of logarithms and their relationship to exponential functions. Introduce the concept of solving equations and the importance of properties of logarithms.
  • One-to-One Property Explanation (5 mins)
    Play the section of the video (0:09 - 0:41) explaining the one-to-one property of logarithms and its connection to the one-to-one property of exponents. Emphasize the condition that the bases of the logarithms must be the same.
  • Example 1: Basic Application (7 mins)
    Play the section of the video (0:42 - 1:08) demonstrating a simple application of the one-to-one property. Pause at key steps to ask students to predict the next step and explain their reasoning. Discuss the importance of checking the solution in the original equation.
  • Example 2: Using Log Properties (10 mins)
    Play the section of the video (1:09 - 1:48) where the quotient property is used before applying the one-to-one property. Discuss how properties of logarithms can simplify equations and make them solvable. Have students actively participate in simplifying the expression before solving.
  • Example 3: Factoring and Extraneous Solutions (8 mins)
    Play the section of the video (1:49 - 2:20) showing an example that requires factoring and leads to extraneous solutions. Emphasize the importance of checking solutions and understanding why some solutions are extraneous (arguments of logs must be positive).
  • Extraneous Solutions Explained (5 mins)
    Play the section of the video (2:21 - end) explaining extraneous solutions. Engage students in a discussion about the domain of logarithmic functions and how it relates to extraneous solutions.
  • Practice Problems (10 mins)
    Provide students with practice problems similar to the examples in the video. Encourage them to work independently or in small groups and to check their solutions. Circulate to provide assistance and answer questions.

Interactive Exercises

  • Whiteboard Challenge
    Present logarithmic equations on the whiteboard and have students race to solve them, explaining their steps as they go. Focus on accuracy and clear explanations.
  • Error Analysis
    Provide worked-out solutions to logarithmic equations with common errors. Have students identify the errors and explain how to correct them.

Discussion Questions

  • Why is it important for the bases of the logarithms to be the same when using the one-to-one property?
  • What are extraneous solutions, and why do they sometimes arise when solving logarithmic equations?
  • How can properties of logarithms help simplify equations before applying the one-to-one property?

Skills Developed

  • Problem-solving
  • Critical thinking
  • Algebraic manipulation
  • Application of logarithmic properties

Multiple Choice Questions

Question 1:

The one-to-one property of logarithms states that if log<sub>b</sub>(x) = log<sub>b</sub>(y), then:

Correct Answer: x = y

Question 2:

Solve for x: log<sub>5</sub>(3x - 2) = log<sub>5</sub>(10)

Correct Answer: x = 4

Question 3:

Which property of logarithms is used to combine log(a) - log(b) into a single logarithm?

Correct Answer: Quotient Property

Question 4:

Solve for x: ln(x) - ln(3) = ln(5)

Correct Answer: x = 15

Question 5:

An extraneous solution is a solution that:

Correct Answer: Does not satisfy the original equation

Question 6:

When solving logarithmic equations, it's important to check for extraneous solutions because the argument of a logarithm must be:

Correct Answer: Positive

Question 7:

Solve for x: log<sub>2</sub>(x<sup>2</sup> - 3) = log<sub>2</sub>(2x)

Correct Answer: x = 3

Question 8:

In the equation from the previous question, is x = -1 an extraneous solution?

Correct Answer: Yes

Question 9:

The natural log (ln) has a base of:

Correct Answer: e

Question 10:

What is the first step to solving: log(x + 2) = log(3x - 4)

Correct Answer: Set the arguments equal to each other

Fill in the Blank Questions

Question 1:

The one-to-one property for logarithms can only be applied when the logarithms have the same ________.

Correct Answer: base

Question 2:

To combine logarithmic terms with subtraction, you can use the ________ property.

Correct Answer: quotient

Question 3:

A solution that does not satisfy the original equation is called an ________ solution.

Correct Answer: extraneous

Question 4:

The argument of a logarithm must always be ________.

Correct Answer: positive

Question 5:

ln(x) represents the logarithm with base ________.

Correct Answer: e

Question 6:

Before applying the one-to-one property, it may be necessary to ________ logarithmic expressions using properties of logarithms.

Correct Answer: simplify

Question 7:

When solving log<sub>b</sub>(x) = log<sub>b</sub>(y), you can conclude that x ________ y.

Correct Answer: equals

Question 8:

If you get a solution that makes the argument of the log negative, then that is an __________ solution.

Correct Answer: extraneous

Question 9:

To solve the equation log<sub>2</sub>(x+3) = log<sub>2</sub>(5), set x+3 equal to _______.

Correct Answer: 5

Question 10:

The one-to-one property allows you to solve equations where the arguments of the log must be _______.

Correct Answer: equal

Educational Standards

A2.A.1.6:Solve common and natural logarithmic equations using the properties of logarithms. A2.A.1.3:Solve one-variable rational equations and check for extraneous solutions.

Teaching Materials

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