Problem 15 - Algebra and Trigonometry

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: Identify the Expression Inside Logarithm

The given logarithmic expression is \( \log_{2}(x(x-1)) \). Notice that the argument of the logarithm is a product of two expressions: \( x \) and \( (x-1) \).

Step 2: Apply the Product Rule of Logarithms

Use the product rule of logarithms, which states \( \log_{b}(MN) = \log_{b}M + \log_{b}N \), to expand the expression. Here, set \( M = x \) and \( N = (x-1) \). Thus, the expression becomes:\[ \log_{2}(x(x-1)) = \log_{2}x + \log_{2}(x-1) \]

Step 3: Write the Final Expanded Expression

The expression has been expanded using the product rule. The final expanded form of the logarithmic expression is:\( \log_{2}x + \log_{2}(x-1) \).

Key Concepts

Product Rule

The product rule is one of the core principles in the logarithmic world. When you encounter a logarithm of a product, like \( \, \log_{b}(MN) \), the product rule allows you to break it down into a simpler expression. The rule can be stated as follows:

• \( \log_{b}(MN) = \log_{b}(M) + \log_{b}(N) \)

This rule is incredibly useful because it turns a potentially complex multiplication inside a logarithmic expression into a simple addition of two separate logs. It simplifies calculations and helps in breaking down complex expressions. By learning this rule, you turn difficult multiplications into simpler additions, making further calculations much easier.

Logarithmic Expressions

Logarithmic expressions are mathematical expressions that involve logarithms. A logarithm answers the question "how many times must we multiply a certain base to get the desired value?" For example, in the expression \( \log_{2}(x(x-1)) \), we're working with logarithms having a base 2. Some key points about exponential expressions include:

• They're often used to solve problems involving exponential growth and decay.
• Understanding them requires knowing basic rules such as product, quotient, and power rules.

In order for you to work with these expressions effectively, it’s essential to understand how to manipulate and break them down using logarithmic laws. This ability to expand complex terms into simpler components is vital, especially in calculus and algebra.

Mathematical Expansion

Mathematical expansion refers to the process of expressing a mathematical expression in an extended form. When you expand an expression, you transform it into a simpler or more comprehensive form. Using our example, \( \log_{2}(x(x-1)) \), the expansion involves applying the product rule, thus changing it into the sum \( \log_{2}(x) + \log_{2}(x-1) \). This makes it easier to work with, whether for calculus purposes or simplifying for solving.The benefits of mathematical expansion include:

• Making complex expressions more manageable.
• Revealing hidden structures or elements in the form of individual terms.
• Helping in further manipulation and solving of the expression.

Mastering expansion techniques allows you to simplify and efficiently handle a variety of mathematical problems.