Unlocking Exponential Equations: The One-to-One Property

Algebra 2 Grades High School 2:32 Video
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Lesson Description

Master the one-to-one property of exponents to solve exponential equations. This lesson provides a clear understanding of the concept with step-by-step examples.

Video Resource

One to One Property for Exponential Equations

Mario's Math Tutoring

Duration: 2:32
Watch on YouTube

Key Concepts

  • One-to-One Property of Exponents
  • Rewriting Numbers with the Same Base
  • Solving Exponential Equations

Learning Objectives

  • Students will be able to state and apply the one-to-one property of exponents.
  • Students will be able to rewrite numbers to have the same base in exponential equations.
  • Students will be able to solve exponential equations using the one-to-one property.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the basic definition of exponents. Then, introduce the one-to-one property: if a^x = a^y, then x = y. Briefly illustrate this property with a simple example like 2^x = 2^3, therefore x=3.
  • Video Presentation (10 mins)
    Play the video 'One to One Property for Exponential Equations' by Mario's Math Tutoring (https://www.youtube.com/watch?v=ABjTZaXl7_s). Encourage students to take notes on the examples provided.
  • Guided Practice (15 mins)
    Work through the examples from the video on the board, pausing to explain each step. Emphasize the process of rewriting numbers to have a common base. Example 1: 3^x = 81 becomes 3^x = 3^4, so x=4. Example 2: 2^(x-2) = 1/16 becomes 2^(x-2) = 2^(-4), so x-2=-4, x=-2. Example 3: e^(x^2) = e^4, so x^2=4, x = +/- 2. Example 4: 16^x = 2 becomes 2^(4x) = 2^1, so 4x=1, x=1/4.
  • Independent Practice (15 mins)
    Assign practice problems where students apply the one-to-one property to solve exponential equations. Include problems where students need to rewrite numbers with common bases. Use the interactive exercises for this section.
  • Wrap-up and Assessment (5 mins)
    Summarize the key concepts of the lesson. Administer a short multiple-choice or fill-in-the-blank quiz to assess student understanding.

Interactive Exercises

  • Practice Problems
    Solve the following exponential equations using the one-to-one property: 1. 5^x = 125 2. 4^(x+1) = 64 3. 3^(2x-1) = 1/9 4. 9^x = 3 5. e^(3x) = e^9

Discussion Questions

  • Why is it important to have the same base when using the one-to-one property of exponents?
  • Can you always rewrite exponential equations to have the same base? What are the limitations?
  • How does the one-to-one property simplify solving exponential equations?

Skills Developed

  • Problem-Solving
  • Algebraic Manipulation
  • Critical Thinking

Multiple Choice Questions

Question 1:

What is the one-to-one property of exponents?

Correct Answer: If a^x = a^y, then x = y

Question 2:

Solve for x: 2^x = 32

Correct Answer: 5

Question 3:

Solve for x: 3^(x-1) = 81

Correct Answer: 5

Question 4:

Solve for x: 5^(2x) = 25

Correct Answer: 1

Question 5:

Solve for x: 4^x = 2

Correct Answer: 0.5

Question 6:

Solve for x: e^(x+2) = e^5

Correct Answer: 3

Question 7:

Solve for x: 10^x = 1000

Correct Answer: 3

Question 8:

Solve for x: 7^(x-3) = 49

Correct Answer: 7

Question 9:

Solve for x: 6^(3x) = 36

Correct Answer: 2/3

Question 10:

Solve for x: 9^(x+1) = 81

Correct Answer: 0

Fill in the Blank Questions

Question 1:

According to the one-to-one property, if a^x = a^y, then x = ____.

Correct Answer: y

Question 2:

To use the one-to-one property, the bases of the exponential expressions must be the ____.

Correct Answer: same

Question 3:

Solve for x: 5^x = 125. x = ____.

Correct Answer: 3

Question 4:

Solve for x: 2^(x+1) = 16. x = ____.

Correct Answer: 3

Question 5:

Solve for x: 49^x = 7. x = ____.

Correct Answer: 0.5

Question 6:

Solve for x: e^(2x) = e^6. x = ____.

Correct Answer: 3

Question 7:

Solve for x: 10^(x-2) = 1000. x = ____.

Correct Answer: 5

Question 8:

Solve for x: 8^(x+1) = 64. x = ____.

Correct Answer: 1

Question 9:

Solve for x: 3^(5x) = 243. x = ____.

Correct Answer: 1

Question 10:

Solve for x: 16^(x-1) = 256. x = ____.

Correct Answer: 3

Educational Standards

A2.A.1.2:Use mathematical models to represent exponential relationships, such as compound interest, depreciation, and population growth. Solve these equations algebraically or graphically (including graphing calculator or other appropriate technology). A2.A.2.5:Rewrite algebraic expressions involving radicals and rational exponents using the properties of exponents.

Teaching Materials

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