Solving Logarithmic Expressions Log₃(b) ⋅ Log_b(27)

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When tackling logarithmic problems, understanding the fundamental properties of logarithms is key. In this comprehensive guide, we will delve into the intricacies of the expression log₃(b) ⋅ log_b(27), breaking it down step by step to reveal its equivalent form. We'll explore the underlying logarithmic principles, providing a clear and concise methodology for solving such equations. This guide is designed to empower you with the knowledge and confidence to navigate the world of logarithms with ease.

Unveiling the Logarithmic Identity

The heart of solving this problem lies in recognizing and applying a crucial logarithmic identity: the change of base formula. This formula allows us to express a logarithm in one base in terms of logarithms in another base. Specifically, the change of base formula states that:

logₐ(x) = logₓ(x) / logₓ(a)

Where 'a' and 'x' are positive real numbers and 'a' is not equal to 1. This identity is a powerful tool in simplifying and solving logarithmic expressions.

Let's start with the given expression:

log₃(b) ⋅ log_b(27)

Our goal is to simplify this expression to one of the given answer choices. To do this, we'll leverage the change of base formula.

Applying the Change of Base Formula

The change of base formula is the cornerstone of simplifying this expression. We can rewrite log_b(27) using base 3. Applying the change of base formula, we get:

log_b(27) = log₃(27) / log₃(b)

This transformation allows us to express both logarithmic terms in terms of base 3, paving the way for simplification.

Now, substitute this back into the original expression:

log₃(b) ⋅ [log₃(27) / log₃(b)]

Notice that log₃(b) appears in both the numerator and denominator, allowing us to cancel them out:

[log₃(b) ⋅ log₃(27)] / log₃(b) = log₃(27)

This simplification brings us closer to the solution. We've effectively eliminated one of the logarithmic terms, leaving us with a single logarithm.

Simplifying log₃(27)

Now we have a much simpler expression: log₃(27). To further simplify this, we need to recognize that 27 is a power of 3. Specifically:

27 = 3³

Therefore, we can rewrite the expression as:

log₃(3³)

Applying another fundamental logarithmic property, we know that the logarithm of a number raised to an exponent is equal to the exponent times the logarithm of the number. In other words:

logₐ(xⁿ) = n ⋅ logₐ(x)

In our case, this translates to:

log₃(3³) = 3 ⋅ log₃(3)

Another key logarithmic identity comes into play here: the logarithm of a number to the same base is always 1.

logₐ(a) = 1

Therefore:

log₃(3) = 1

Substituting this back into our expression, we get:

3 ⋅ log₃(3) = 3 ⋅ 1 = 3

Thus, the simplified form of the original expression is 3.

Conclusion: The Equivalent Expression

By meticulously applying the change of base formula and fundamental logarithmic properties, we've successfully simplified the expression log₃(b) ⋅ log_b(27) to 3. This corresponds to answer choice A.

Therefore, the correct answer is:

A. 3

This exercise demonstrates the power of logarithmic identities in simplifying complex expressions. By mastering these identities and practicing their application, you can confidently tackle a wide range of logarithmic problems.

To further enhance your understanding and problem-solving skills in logarithms, consider these additional tips:

  • Memorize the Key Logarithmic Identities: Familiarity with identities like the change of base formula, the product rule, the quotient rule, and the power rule is essential for efficient problem-solving.
  • Practice Regularly: Consistent practice is crucial for mastering logarithmic concepts and techniques. Work through a variety of problems to solidify your understanding.
  • Rewrite Expressions: When faced with a complex logarithmic expression, try rewriting it in a simpler form using logarithmic identities. This often reveals hidden relationships and simplifies the problem.
  • Think About the Base: Pay close attention to the base of the logarithm. If possible, try to express all logarithms in the same base to facilitate simplification.
  • Check Your Answers: Always check your answers to ensure they are valid and make sense in the context of the original problem.

By incorporating these tips into your problem-solving approach, you'll be well-equipped to conquer even the most challenging logarithmic equations.

The step-by-step simplification process clearly demonstrates why answer choice A, 3, is the correct equivalent of the given expression. The strategic application of the change of base formula and logarithmic properties allows us to systematically reduce the complexity of the expression until we arrive at the numerical value of 3. This methodical approach highlights the importance of understanding and applying these fundamental concepts in solving logarithmic problems.

When working with logarithms, it's easy to fall into common traps. Here are some mistakes to avoid:

  • Incorrectly Applying the Change of Base Formula: Ensure you are using the formula correctly, with the correct arguments in the numerator and denominator.
  • Forgetting Logarithmic Identities: A lack of familiarity with key identities can hinder simplification efforts.
  • Ignoring the Base: Failing to consider the base of the logarithm can lead to errors in simplification.
  • Incorrectly Simplifying Exponents: Remember the rules for exponents when dealing with logarithmic expressions.
  • Not Checking for Extraneous Solutions: In some cases, solutions obtained may not be valid in the original equation. Always check your answers.

By being aware of these common pitfalls, you can minimize errors and improve your accuracy in solving logarithmic problems.

Logarithms are not just abstract mathematical concepts; they have a wide range of real-world applications. From measuring the intensity of earthquakes (the Richter scale) to determining the acidity of a solution (pH scale), logarithms play a crucial role in various fields of science and engineering. Understanding logarithms is essential for anyone pursuing studies in these areas.

In computer science, logarithms are used in analyzing the efficiency of algorithms. In finance, they are used to calculate compound interest and analyze investment growth. The applications of logarithms are vast and varied, highlighting their importance in both theoretical and practical contexts.

By mastering the fundamentals of logarithms, you are not only enhancing your mathematical skills but also equipping yourself with a powerful tool for understanding and solving problems in a wide range of disciplines.

Logarithms may seem daunting at first, but with a solid understanding of their properties and consistent practice, they become manageable and even enjoyable. This guide has provided a detailed walkthrough of solving a specific logarithmic problem, along with valuable tips and insights to help you on your logarithmic journey. Remember to focus on understanding the underlying principles, practice regularly, and don't be afraid to seek help when needed. With dedication and perseverance, you can unlock the power of logarithms and excel in your mathematical endeavors.