Problem 22 - Algebra 2 and Trigonometry

Question

a. Expand each expression as a difference, sum, and/or multiple of logarithms. b. Find the value of each expression. \(\frac{1}{2} \log _{3} 3(243)\)

Step-by-Step Solution

Answer

The expanded expression is \(\frac{1}{2}(1 + 5)\), and the value is 3.

Step 1: Recognize the Expression Format

The given expression is \(\frac{1}{2} \log_{3} (3 \times 243)\). This is a product inside the logarithm, which we can expand using logarithmic properties.

Step 2: Apply the Product Rule for Logarithms

The product rule for logarithms says \(\log_b (xy) = \log_b x + \log_b y\). Thus, \(\log_{3} (3 \times 243) = \log_{3} 3 + \log_{3} 243\).

Step 3: Calculate Each Logarithm

We know \(\log_{3} 3 = 1\) because any number log base to the same number equals 1. Next, we find \(\log_{3} 243\). Since \(243 = 3^5\), we use \(\log_b (a^n) = n \cdot \log_b a\), getting \(\log_3 243 = 5 \cdot \log_3 3 = 5 \cdot 1 = 5\).

Step 4: Sum the Logarithms

From the previous calculations, \(\log_{3} (3 \times 243) = 1 + 5 = 6\).

Step 5: Apply the Coefficient

The expression starts with \(\frac{1}{2} \cdot \log_{3} (3 \times 243)\), which means we multiply our result by \(\frac{1}{2}\): \(\frac{1}{2} \cdot 6 = 3\).

Key Concepts

Product Rule for Logarithms

The product rule for logarithms is a powerful tool when dealing with logarithmic expressions involving multiplication. This rule states that for any base \( b \), the logarithm of a product is equal to the sum of the logarithms of its factors. Mathematically, the rule is represented as:

• \( \log_b(xy) = \log_b x + \log_b y \)

To apply this to the expression \( \log_{3} (3 \times 243) \), we break it down using the product rule. By representing this multiplication inside the logarithm as a sum:

• \( \log_{3} 3 + \log_{3} 243 \)

This step simplifies complex expressions and makes calculations more manageable. Remember, using the product rule effectively requires familiarity with breaking down numbers into smaller factors, especially when working with logarithms.

Logarithmic Properties

Logarithmic properties are essential when working to simplify or expand expressions involving logarithms. These properties include the product rule, the power rule, and the quotient rule. They form the foundation for manipulating logarithmic expressions. One crucial property we used is the power rule, represented as:

• \( \log_b (a^n) = n \cdot \log_b a \)

This allowed us to transform \( \log_{3} 243 \). Observing that \( 243 = 3^5 \), we applied this rule:

• \( \log_3 243 = 5 \cdot \log_3 3 \)

Understanding these properties helps in quickly identifying how to simplify or expand logarithmic expressions, making them easier to solve. As you dive into logarithmic exercises, periodically review these properties to reinforce your knowledge.

Calculating Logarithms

Calculating logarithms can seem daunting, but with the right approach, it becomes straightforward. Start by recognizing simple logarithmic values, such as \( \log_3 3 = 1 \). This is known because a logarithm indicates the power needed for the base to reach a number: here, \( 3^1 = 3 \). Additionally, when factoring numbers—like 243 into \( 3^5 \)—and employing logarithmic properties, you can simplify calculations. After breaking down the logarithmic expression using properties, we calculated:

• \( \log_3 (3 \times 243) = 6 \)
• Multiplying by any coefficient, like \( \frac{1}{2} \), if present gives the final answer.

Thus, for the given problem, the final step was:

• \( \frac{1}{2} \cdot 6 = 3 \)

Mastering these techniques involves practice and applying these principles regularly, improving your efficiency and confidence in calculating logarithms.