Problem 21 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

\( \log_2 (x) + \log_2 (x-1) \)

Step 1: Recognize the Product Rule

The expression inside the logarithm is \( x(x-1) \), which is the product of two terms: \( x \) and \( x-1 \). According to the product rule of logarithms, \( \log_b (MN) = \log_b M + \log_b N \).

Step 2: Apply the Product Rule

Using the product rule, split the logarithm into two separate logarithms: \( \log_2 (x) + \log_2 (x-1) \).

Step 3: Confirm the Expansion

Verify that the expression \( \log_2 (x(x-1)) \) is correctly expanded to \( \log_2 (x) + \log_2 (x-1) \) using the product law of logarithms. Each part of the product inside the logarithm is now individually logged.

Key Concepts

Product Rule

When working with logarithms, the product rule is a helpful tool to know. This rule states that the logarithm of a product is equal to the sum of the logarithms of its factors. In simpler terms, if you have a logarithm with two multiplied terms inside, you can break it down into separate logarithms by adding them together. This is written as:\[\log_b (MN) = \log_b M + \log_b N\]For example, if you have an expression like \( \log_2(x(x-1)) \), you can use the product rule to expand it into \( \log_2(x) + \log_2(x-1) \). This method simplifies complex logarithmic expressions, making them easier to manage and solve. By practicing the product rule, you'll become more adept at navigating through logarithm problems in algebra and beyond.

Logarithmic Expansion

Logarithmic expansion refers to the process of breaking down complex logarithmic expressions into simpler, more manageable parts. When we apply the laws of logarithms, particularly the product rule, we perform a logarithmic expansion.In the given problem \( \log_2(x(x-1)) \), the expansion involves recognizing and applying the product rule to separate the terms inside the logarithm. This transforms the expression into \( \log_2(x) + \log_2(x-1) \). Using this technique makes it easy to handle expressions with larger numbers of factors.Logarithmic expansion is extremely useful in various fields such as mathematics, engineering, and science, as it aids in simplifying and solving complex problems that involve multiplication or division inside a logarithmic term. By mastering this concept, you'll be able to effortlessly expand and work with much more complex expressions in your studies.

Logarithm Properties

Logarithms possess a set of fundamental properties that can greatly simplify your mathematical computations and problem-solving skills. These properties include the product rule, which you've already learned about, as well as others like the quotient and power rules.

• **Product Rule**: Already explained, \( \log_b (MN) = \log_b M + \log_b N \), helps in expanding products.
• **Quotient Rule**: Allows the division inside a log to be split, \( \log_b (\frac{M}{N}) = \log_b M - \log_b N \).
• **Power Rule**: Lets us deal with an exponent inside the log, \( \log_b (M^k) = k \cdot \log_b M \).

Understanding these properties can change how you approach problems. Each property provides a strategy for simplifying expressions and solving equations more efficiently.For instance, knowing these properties equips you with the tools to deconstruct and simplify logarithmic equations, making complex ones much more approachable. Thus, they play a crucial role in both theoretical and practical applications, from solving basic algebraic equations to higher-level calculus.