Problem 22 - Algebra and Trigonometry

Question

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log _{5} \frac{x}{2} $$

Step-by-Step Solution

Answer

\( \log_5 \frac{x}{2} = \log_5 x - \log_5 2 \).

Step 1: Identify the Logarithmic Rule

To expand the expression \( \log_5 \frac{x}{2} \), we need to identify the logarithmic rule that applies. Here, we can use the quotient rule for logarithms, which states that \( \log_b \frac{M}{N} = \log_b M - \log_b N \).

Step 2: Apply the Quotient Rule

Apply the quotient rule to the logarithmic expression: \( \log_5 \frac{x}{2} = \log_5 x - \log_5 2 \). This separates the division inside the logarithm into a difference of two logarithms with the same base.

Key Concepts

Quotient Rule for Logarithms

The quotient rule for logarithms is a very handy tool when dealing with the division within a logarithmic expression. It simplifies expressions where a division is involved by turning them into a subtraction of two separate logs. The rule can be stated as follows: if you have an expression like \( \log_b \frac{M}{N} \), you can rewrite it as \( \log_b M - \log_b N \). Essentially, the quotient rule implies that the logarithm of a division is the difference of the logarithms.

• Say you have a logarithm of a fraction, such as \( \log_5 \frac{x}{2} \).
• Applying the quotient rule, this becomes \( \log_5 x - \log_5 2 \).

This rule is powerful because it can simplify the process of breaking down complex logarithmic expressions. Always look for opportunities to apply this rule when you see a fraction inside a logarithm.

Laws of Logarithms

Logarithms have several important rules, called the laws of logarithms, which help simplify and manipulate expressions for easier calculation. These laws are extremely useful when expanding or condensing logarithmic expressions. Here are the primary laws you need to remember:

• **Product Rule**: \( \log_b (MN) = \log_b M + \log_b N \)
• **Quotient Rule**: \( \log_b \frac{M}{N} = \log_b M - \log_b N \)
• **Power Rule**: \( \log_b (M^p) = p \cdot \log_b M \)

Using these laws can transform complicated logarithmic expressions into simpler, more manageable pieces. In the case of \( \log_5 \frac{x}{2} \), the quotient rule is key. When you see an expression you need to expand, inspect it to see which law applies.

Logarithmic Expressions

Logarithmic expressions are used frequently in mathematics to simplify calculations involving exponential numbers. A logarithmic expression usually involves a base and a number, when expanded, often reveals underlying arithmetic relationships.

Logarithmic expressions like \( \log_b \frac{x}{2} \) indicate how many times a base, here 5, must raise itself to reach another number, represented by the numerator when a fraction is involved. When you expand such expressions using rules like the quotient rule, you break down the division into two simpler expressions: \( \log_5 x - \log_5 2 \).

• This breakdown makes it easier to compute or further manipulate the terms separately.
• Expanded forms are beneficial for solving equations, integrating functions, and simplifying complex expressions.

Grasping how to expand and manipulate logarithmic expressions opens the door to tackling more advanced mathematical problems with confidence.