Unlock Logarithms: Mastering the Change of Base Formula

PreAlgebra Grades High School 2:05 Video
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Lesson Description

Learn how to effectively use the Change of Base Formula to evaluate logarithms with any base, enhancing your problem-solving skills in PreCalculus.

Video Resource

Logarithms Change of Base Formula

Mario's Math Tutoring

Duration: 2:05
Watch on YouTube

Key Concepts

  • Logarithms
  • Change of Base Formula
  • Common Logarithms (Base 10)
  • Natural Logarithms (Base e)

Learning Objectives

  • Understand the purpose and application of the Change of Base Formula.
  • Apply the Change of Base Formula to evaluate logarithms with different bases using common and natural logarithms.
  • Accurately compute logarithms using the Change of Base Formula and a calculator.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of a logarithm and its relationship to exponential functions. Briefly discuss the limitations of some calculators in evaluating logarithms with arbitrary bases and introduce the need for the Change of Base Formula.
  • Introducing the Change of Base Formula (5 mins)
    Present the Change of Base Formula: log_b(x) = log_c(x) / log_c(b). Emphasize that 'c' can be any valid base, but base 10 (common log) and base e (natural log) are most commonly used due to calculator functionality. Explain how x goes in the numerator and b in the denominator.
  • Example 1: Evaluating Log_2(10) (10 mins)
    Work through the example from the video: log_2(10). Demonstrate how to apply the formula using both common logarithms (log_10(10) / log_10(2)) and natural logarithms (ln(10) / ln(2)). Use a calculator to evaluate both expressions and show that they yield the same result. Stress that log_10(10)/log_10(2) is NOT equal to log_10(5).
  • Example 2: Evaluating Log_3(7) (10 mins)
    Work through the second example: log_3(7). Again, demonstrate the use of both common logarithms (log_10(7) / log_10(3)) and natural logarithms (ln(7) / ln(3)). Use a calculator to evaluate and verify the results.
  • Practice Problems (10 mins)
    Provide students with several practice problems to apply the Change of Base Formula. Encourage them to use both common and natural logarithms and compare their results. Examples: log_5(20), log_4(12), log_7(3).
  • Wrap-up and Q&A (5 mins)
    Summarize the key concepts of the Change of Base Formula and its application. Answer any remaining questions from students.

Interactive Exercises

  • Calculator Challenge
    Divide students into groups and assign each group a logarithm with an uncommon base (e.g., log_6(15)). Have them use the Change of Base Formula and calculators to find the value. The first group to get the correct answer wins.

Discussion Questions

  • Why is the Change of Base Formula necessary?
  • What are the advantages of using common or natural logarithms when applying the Change of Base Formula?
  • How does the Change of Base Formula relate to the properties of logarithms?

Skills Developed

  • Applying mathematical formulas
  • Using calculators for logarithmic calculations
  • Problem-solving in mathematics

Multiple Choice Questions

Question 1:

What is the Change of Base Formula?

Correct Answer: log_b(x) = log_c(x) / log_c(b)

Question 2:

In the Change of Base Formula, what is the most common choice for the new base 'c'?

Correct Answer: 10 or e

Question 3:

Using the Change of Base Formula, log_5(12) can be expressed as:

Correct Answer: log(12) / log(5)

Question 4:

Why is the Change of Base Formula useful?

Correct Answer: It allows calculators to compute logarithms with any base.

Question 5:

Which of the following is equivalent to ln(15) / ln(3)?

Correct Answer: log_3(15)

Question 6:

If you calculate log_10(20) / log_10(4) using the Change of Base Formula, what logarithm are you finding?

Correct Answer: log_4(20)

Question 7:

What is the value of log_8(16) after applying the change of base formula and simplifying (using base 2)?

Correct Answer: 4/3

Question 8:

log_a(b) * log_b(a) is equal to:

Correct Answer: 1

Question 9:

The Change of Base formula can be applied to change a base to:

Correct Answer: Any valid base

Question 10:

Which expression is equal to log_x(y)?

Correct Answer: ln(y) / ln(x)

Fill in the Blank Questions

Question 1:

The Change of Base Formula states that log_b(x) = log_c(x) / log_c(__{blank}__).

Correct Answer: b

Question 2:

The common logarithm uses a base of __{blank}__.

Correct Answer: 10

Question 3:

The natural logarithm uses a base of __{blank}__.

Correct Answer: e

Question 4:

To evaluate log_7(15) using the Change of Base Formula with natural logarithms, you would calculate ln(15) / ln(__{blank}__).

Correct Answer: 7

Question 5:

log_a(a) equals __{blank}__.

Correct Answer: 1

Question 6:

Using the change of base formula, log_3(x) = log(x)/log(__{blank}__).

Correct Answer: 3

Question 7:

If you want to change log_5(8) to a logarithm with base 2, the change of base formula would give you log_2(8) / log_2(__{blank}__).

Correct Answer: 5

Question 8:

log_b(1) = __{blank}__ for any valid base b.

Correct Answer: 0

Question 9:

The argument 'x' in log_b(x) always goes in the __{blank}__ of the Change of Base Formula.

Correct Answer: numerator

Question 10:

When using a calculator, the most common bases available for logarithms are base 10 (log) and base e (__{blank}__).

Correct Answer: ln

Educational Standards

PC.F.3.1:[Objective] Predict solutions involving functions that are quadratic, polynomial of higher order, rational, exponential, and logarithmic. PC.F.3.3:[Objective] Algebraically verify solutions involving functions that are quadratic, polynomial of higher order, rational, exponential, and logarithmic.

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