Conquering Logarithmic Equations: Mastering Different Bases

Algebra 2 Grades High School 2:50 Video
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Lesson Description

Learn to solve complex logarithmic equations with different bases using properties of logarithms and inverse operations. This lesson provides a step-by-step approach to isolating variables and simplifying expressions.

Video Resource

Solving (Challenging) Log Equations Different Bases

Mario's Math Tutoring

Duration: 2:50
Watch on YouTube

Key Concepts

  • Properties of Logarithms (Product Rule, Power Rule)
  • Inverse Relationship between Exponential and Logarithmic Functions
  • Changing Bases of Logarithms (implicitly used)

Learning Objectives

  • Students will be able to apply properties of logarithms to expand and simplify logarithmic expressions.
  • Students will be able to solve logarithmic equations with different bases by strategically applying inverse operations and algebraic manipulation.
  • Students will be able to express solutions in exact form and approximate decimal form (using a calculator).

Educator Instructions

  • Introduction (5 mins)
    Briefly review the definition of logarithms and the inverse relationship between exponential and logarithmic functions. Highlight the properties of logarithms (product rule, power rule) that will be used in the lesson.
  • Video Demonstration (10 mins)
    Play the video 'Solving (Challenging) Log Equations Different Bases' by Mario's Math Tutoring. Encourage students to take notes on the steps involved in solving the example problem.
  • Step-by-Step Breakdown (15 mins)
    Go through the video example step-by-step, pausing to explain each step in detail. Emphasize the reasoning behind choosing to take the log base 2 of both sides and how the properties of logarithms are applied. Show how the power rule allows bringing the exponent in front of the logarithm. Clarify how the x is factored out of each term to be solved for. Stress the importance of expressing the final answer in exact form and how to use a calculator to find a decimal approximation.
  • Guided Practice (15 mins)
    Present a similar logarithmic equation with different bases and guide students through the solution process. Provide prompts and hints as needed. Encourage students to work in pairs or small groups.
  • Independent Practice (10 mins)
    Assign a few similar problems for students to solve independently. Provide answers for self-checking.

Interactive Exercises

  • Property Matching
    Provide a list of logarithmic expressions and a list of corresponding simplified expressions using properties of logarithms. Have students match the expressions correctly.
  • Error Analysis
    Present a worked-out solution to a logarithmic equation with a common error. Have students identify the error and correct it.

Discussion Questions

  • Why is it important to perform the same operation on both sides of an equation?
  • How do the properties of logarithms help simplify complex logarithmic expressions?
  • What strategies can you use to decide which base to use when taking the logarithm of both sides of an equation?

Skills Developed

  • Problem-solving
  • Algebraic manipulation
  • Application of Logarithmic Properties

Multiple Choice Questions

Question 1:

Which property of logarithms allows you to rewrite log_b(xy) as log_b(x) + log_b(y)?

Correct Answer: Product Rule

Question 2:

Which property of logarithms allows you to rewrite log_b(x^p) as p*log_b(x)?

Correct Answer: Power Rule

Question 3:

Solve for x: 2^(x+1) = 3^(x). Which of the following is the first step to solve the equation?

Correct Answer: Take log base 2 of both sides

Question 4:

If you have log base 5 of 25, what does that simplify to?

Correct Answer: 2

Question 5:

When solving an equation with logs of different bases, what is a key strategy to simplify?

Correct Answer: Taking the logarithm of both sides with the same base

Question 6:

What is the inverse operation of taking the logarithm of a number?

Correct Answer: Exponentiating

Question 7:

Simplify: log_2(8) + log_2(4)

Correct Answer: 5

Question 8:

Solve for x: log_3(x) = 2

Correct Answer: x = 9

Question 9:

Which of the following is equivalent to log_b(x/y)?

Correct Answer: log_b(x) - log_b(y)

Question 10:

If you factor an 'x' out of the equation x - xlog_2(3), what is left inside the parentheses?

Correct Answer: 1 - log_2(3)

Fill in the Blank Questions

Question 1:

The property of logarithms that allows you to expand log_b(xy) is called the __________ Rule.

Correct Answer: Product

Question 2:

The inverse operation of an exponential function is a __________ function.

Correct Answer: logarithmic

Question 3:

When solving logarithmic equations, it is important to __________ the same operation to both sides of the equation.

Correct Answer: apply

Question 4:

log_b(1) is always equal to __________.

Correct Answer: 0

Question 5:

The property of logarithms that allows you to bring an exponent down is called the __________ Rule.

Correct Answer: Power

Question 6:

If 5^x = 7, then x = log_5(__________).

Correct Answer: 7

Question 7:

The expression log_b(x) - log_b(y) can be simplified to log_b(__________).

Correct Answer: x/y

Question 8:

When solving an equation with variables in the exponent, taking the __________ of both sides is a common first step.

Correct Answer: logarithm

Question 9:

In the expression log_b(a), 'b' is referred to as the __________ of the logarithm.

Correct Answer: base

Question 10:

When you have an x in multiple terms you should __________ it out to isolate the variable.

Correct Answer: factor

Educational Standards

A2.A.1.6:Solve common and natural logarithmic equations using the properties of logarithms. A2.A.2:Generate and evaluate equivalent algebraic expressions and equations using various strategies. A2.F.1.2:Identify the parent forms of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [𝑓(𝑥 + 𝑐), 𝑓(𝑥) + 𝑐, 𝑓(𝑐𝑥), and 𝑐𝑓(𝑥)] algebraically and graphically.

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