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Chapter 4: Problem 15

Use the Laws of Logarithms to expand the expression. $$\log _{2}(x(x-1))$$

Short Answer

Expert verified
\( \log_{2}(x) + \log_{2}(x-1) \)

Step by step solution

01

Identify the expression inside the logarithm

The expression given is \( \log_{2}(x(x-1)) \). Inside the logarithm, we have a product of two terms: \( x \) and \( (x-1) \). The task is to expand this logarithmic expression using logarithmic properties.
02

Apply the Product Rule of Logarithms

The Product Rule states that the logarithm of a product is the sum of the logarithms: \( \log_{b}(MN) = \log_{b}(M) + \log_{b}(N) \). Applying this to our expression, we get: \( \log_{2}(x(x-1)) = \log_{2}(x) + \log_{2}(x-1) \).
03

Simplify the Expression

The expression \( \log_{2}(x) + \log_{2}(x-1) \) is already in its simplest expanded form. We have successfully applied the product rule to expand the logarithmic expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithm Expansion
Expanding a logarithm simplifies the expression, making it easier to handle in algebraic operations. This process uses logarithmic identities and rules. In our context, we started with \( \log_{2}(x(x-1)) \), an expression where the inner content is a product of \( x \) and \( (x-1) \). To expand this expression, we applied specific logarithmic rules. Expansion converts a compound expression into a sum. This is essential in breaking down complex logarithmic statements for simplification or integration into larger equations. Always start by identifying the components inside the logarithm. Think of expansion as a zoom-out operation that reveals simpler parts of a bigger picture.
Product Rule of Logarithms
The Product Rule of Logarithms is a fundamental property that allows you to separate a logarithm of a product into the sum of logarithms. When you have an expression in the form of \( \log_{b}(MN) \), you can rewrite it as \( \log_{b}(M) + \log_{b}(N) \).In practical terms, if you're faced with \( \log_{2}(x(x-1)) \), break it down using the rule:
  • Identify the product: In this exercise, it's \( x(x-1) \).
  • Apply the rule: \( \log_{2}(x(x-1)) = \log_{2}(x) + \log_{2}(x-1) \).
Simplifying expressions in logarithms becomes straightforward when you utilize this rule. It helps in reducing the complexity by transforming products inside logs into sums, allowing each part to be individually addressed or calculated.
Logarithmic Properties
Understanding the properties of logarithms is crucial for manipulating and solving logarithmic equations effectively. With the property spotlight on the Product Rule, it's part of a wider set of rules including the Quotient Rule and Power Rule:
  • Quotient Rule: \( \log_{b}(\frac{M}{N}) = \log_{b}(M) - \log_{b}(N) \).
  • Power Rule: \( \log_{b}(M^n) = n \cdot \log_{b}(M) \).
These properties simplify complex expressions, turning them into manageable algebraic pieces. For any logarithmic expression, a foundational understanding of these properties allows you to break down equations efficiently, making solutions more accessible.

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