Problem 71 - Algebra A Combined Function

Question

Solve. Which of the following is the correct way to rewrite \(\log _{3} \frac{14}{11} ?\) a. \(\frac{\log _{3} 14}{\log _{3} 11}\) b. \(\log _{3} 14-\log _{3} 11\) c. \(\log _{3}(14-11)\) d. \(\log _{3} 154\)

Step-by-Step Solution

Answer

The correct answer is b. \( \log_{3} 14 - \log_{3} 11 \).

Step 1: Review the Logarithmic Property

Recall the logarithmic property: the logarithm of a quotient is the difference of the logarithms. Specifically, \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \).

Step 2: Apply the Property to the Given Expression

Use the quotient property to rewrite the given expression, \( \log_{3} \frac{14}{11} \), as \( \log_{3} 14 - \log_{3} 11 \).

Step 3: Compare With the Given Choices

Examine the options provided in the problem. Option (b) is \( \log_{3} 14 - \log_{3} 11 \), which matches our rewritten expression from Step 2.

Key Concepts

Logarithm of a Quotient

Understanding the concept of the logarithm of a quotient is fundamental in solving various logarithmic equations. The logarithm of a quotient is expressed as the difference between two logarithms. This means when you have an expression of the form \( \log_b \left( \frac{M}{N} \right) \), it can be rewritten as \( \log_b M - \log_b N \). This property is exceptionally useful in simplifying expressions and solving logarithmic equations by breaking down more complex fractions into manageable parts. The property relies on the mathematics of logarithms which transforms division inside the logarithm into subtraction of logs outside the logarithm. This transformation is backed by the fact that logarithms, being the inverse of exponentiation, naturally decompose multiplication into addition and division into subtraction.

Logarithm Rules

Logarithm rules, also known as logarithmic identities, are crucial tools that help solve mathematical problems involving logs. Among the most important rules are:

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

These rules mean that logarithms can manipulate and break down more complicated expressions into simpler forms using addition, subtraction, and multiplication respectively. The quotient rule, in particular, is applied in problems such as \( \log_{3} \frac{14}{11} \), allowing us to convert complex fractions under a log into a simpler, more digestible format. These rules form the backbone for further logarithmic manipulations and analyses.

Math Problem Solving

Solving math problems effectively requires understanding and applying the right mathematical concepts and rules. For logarithmic problems, identifying which properties or rules to apply is the first crucial step.Consider the problem \( \log_{3} \frac{14}{11} \). By identifying it as a logarithm of a quotient, we can use the corresponding rule to transform and simplify it as \( \log_{3} 14 - \log_{3} 11 \). Upon reviewing the options for solving, you'll see that option (b), \( \log_{3} 14 - \log_{3} 11 \), matches our transformed expression.Effective math problem-solving is about recognizing patterns and properties, practicing frequently, and debugging solutions until the correct one emerges. With logarithms, a firm grasp on core logarithmic rules and properties equips you with the tools necessary to solve typical log-related questions comfortably.