Logarithms to the Rescue: Solving Exponential Equations

Algebra 2 Grades High School 1:46 Video
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Lesson Description

This lesson plan teaches students how to solve exponential equations using logarithms, focusing on the inverse relationship between exponential functions and logarithms and the change of base formula. Students will learn to apply these concepts to solve problems where bases cannot be easily matched.

Video Resource

Solving Exponential Equations

Mario's Math Tutoring

Duration: 1:46
Watch on YouTube

Key Concepts

  • Inverse relationship between exponential and logarithmic functions
  • Logarithms as a tool to solve exponential equations
  • Change of base formula for logarithms

Learning Objectives

  • Students will be able to solve exponential equations using logarithms.
  • Students will be able to apply the change of base formula to evaluate logarithms with different bases.
  • Students will be able to identify when logarithms are necessary to solve exponential equations.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of exponential functions and the concept of inverse functions. Briefly discuss situations where it is difficult to solve exponential equations by simply matching the bases. Introduce the idea that logarithms can be used as the inverse operation to solve these types of equations.
  • Video Viewing (7 mins)
    Play the video 'Solving Exponential Equations' by Mario's Math Tutoring. Instruct students to take notes on the steps involved in solving the example problem.
  • Guided Practice (15 mins)
    Work through a similar example problem on the board, emphasizing each step. Explain why taking the logarithm of both sides is a valid operation. Show how to use the change of base formula when necessary. Encourage student participation by asking guiding questions at each step.
  • Independent Practice (15 mins)
    Provide students with a worksheet containing several exponential equations to solve using logarithms. Include problems where they need to use the change of base formula. Circulate the room to provide assistance as needed.
  • Wrap-up and Discussion (8 mins)
    Review the solutions to the independent practice problems. Address any remaining questions or misconceptions. Summarize the key steps in solving exponential equations using logarithms.

Interactive Exercises

  • Online Logarithm Calculator
    Use an online logarithm calculator to demonstrate how to evaluate logarithms with different bases. This allows students to visually confirm the solutions they find using the change of base formula.
  • Group Problem Solving
    Divide the class into small groups and assign each group a challenging exponential equation to solve. Have each group present their solution to the class, explaining their reasoning and approach.

Discussion Questions

  • Why can't we always solve exponential equations by making the bases the same?
  • What is the relationship between exponential functions and logarithms?
  • When is it necessary to use the change of base formula, and why does it work?

Skills Developed

  • Algebraic manipulation
  • Problem-solving
  • Application of mathematical concepts

Multiple Choice Questions

Question 1:

What is the first step in solving an exponential equation where the bases cannot be easily matched?

Correct Answer: Take the logarithm of both sides

Question 2:

Which of the following is the inverse function of y = a^x?

Correct Answer: y = log_a(x)

Question 3:

The change of base formula allows you to evaluate a logarithm with any base using:

Correct Answer: Both B and C

Question 4:

Solve for x: 3^x = 20

Correct Answer: x = log_3(20)

Question 5:

When solving an exponential equation using logarithms, what property of logarithms is being used?

Correct Answer: Inverse Property

Question 6:

What is the purpose of using logarithms to solve exponential equations?

Correct Answer: To isolate the variable in the exponent

Question 7:

If log_b(a) = c, then b^c = ?

Correct Answer: a

Question 8:

The expression log_5(25) is equal to:

Correct Answer: 2

Question 9:

Which expression is equivalent to log_2(8)/log_2(4)?

Correct Answer: log_2(2)

Question 10:

If the solution to an exponential equation is x = log_3(10) - 1, which of the following represents the approximate decimal value for x?

Correct Answer: 1.0959

Fill in the Blank Questions

Question 1:

Logarithms are the _________ of exponential functions.

Correct Answer: inverses

Question 2:

The change of base formula allows you to rewrite a logarithm in terms of another ________.

Correct Answer: base

Question 3:

To solve 5^x = 30, you would take the _________ of both sides.

Correct Answer: logarithm

Question 4:

log_b(1) = _______ for any base b.

Correct Answer: 0

Question 5:

log_b(b) = _______ for any base b.

Correct Answer: 1

Question 6:

The equation log_2(x) = 3 is equivalent to 2^3 = _______.

Correct Answer: x

Question 7:

The function f(x) = 2^x represents exponential _______.

Correct Answer: growth

Question 8:

The graph of an exponential function has a horizontal _________.

Correct Answer: asymptote

Question 9:

If you cannot easily match the bases, use ________ to solve.

Correct Answer: logarithms

Question 10:

The change of base formula is log_b(a) = log(a) / log(___).

Correct Answer: b

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.1.6:Solve common and natural logarithmic equations using the properties of logarithms. A2.F.1.4:Graph exponential and logarithmic functions. Identify the domain, range, asymptotes, and x- and y-intercepts using various methods and tools that may include calculators or other appropriate technology. Recognize exponential decay and growth graphically and algebraically.

Teaching Materials

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