Problem 104 - Elementary and Intermediate Algebra

Question

Which pair of expressions in each list are equivalent? a. \(\log _{6} \frac{7}{9}\) b. \(\frac{\log _{6} 7}{\log _{6} 9}\) c. \(\log _{6} 7-\log _{6} 9\)

Step-by-Step Solution

Answer

Expressions (a) and (c) are equivalent.

Step 1: Understand Logarithmic Properties

To determine equivalence, recall a key property of logarithms: the quotient rule. It states that the logarithm of a quotient is equal to the difference of the logarithms: \[ \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \] This central property helps us identify equivalent expressions.

Step 2: Analyze Expression (a)

Expression (a) is \( \log_6 \frac{7}{9} \). According to the quotient rule outlined in Step 1, this can be rewritten as:\[ \log_6 \frac{7}{9} = \log_6 7 - \log_6 9 \]

Step 3: Analyze Expression (b)

Expression (b) is \( \frac{\log_6 7}{\log_6 9} \). This expression represents the ratio of two logarithms and doesn’t initially suggest the application of any logarithmic properties directly related to the form from Step 1.

Step 4: Analyze Expression (c)

Expression (c) is \( \log_6 7 - \log_6 9 \). Compare this to Step 2, where the application of the quotient rule revealed that this expression is identical to Expression (a):\[ \log_6 7 - \log_6 9 = \log_6 \frac{7}{9} \]

Step 5: Identify Equivalent Pair

From Steps 2 and 4, we have found that Expressions (a) \( \log_6 \frac{7}{9} \) and (c) \( \log_6 7 - \log_6 9 \) are equivalent as they both simplify to the same expression using logarithmic properties.

Key Concepts

Equivalent Expressions

Equivalent expressions are mathematical expressions that yield the same value when evaluated, even if they look different. This concept is pivotal in algebra and calculus as it allows us to transform complex expressions into more manageable forms.
For instance, consider the expressions involving logarithms. While they might appear different at first glance, they can be simplified or rewritten to reveal their equivalence. In the original exercise, expressions \( \log_6 \frac{7}{9} \) and \( \log_6 7 - \log_6 9 \) look distinct but are equivalent. This transformation often involves applying the rules of logarithms.

• Rewriting logarithms using rules reveals equivalent expressions.
• Recognizing equivalent forms helps simplify problem-solving.

Logarithmic Properties

Logarithmic properties provide the tools needed to manipulate and simplify logarithmic expressions. The three main rules are the product rule, quotient rule, and power rule. These properties enable us to transform and evaluate logarithmic expressions effectively.

Common Logarithmic Properties

• Product Rule: \( \log_b(MN) = \log_b M + \log_b N \) - When multiplying inside a log, it turns into addition.
• Quotient Rule: \( \log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N \) - Division inside a log translates to subtraction.
• Power Rule: \( \log_b(M^k) = k \cdot \log_b M \) - A power inside the log can be brought in front as a multiplier.

Understanding these properties is crucial, as they serve as the basis for determining equivalence between logarithmic expressions. They are like algebra's Swiss Army knife, providing various tools to adjust and simplify expressions.

Quotient Rule

The quotient rule in logarithms is a fundamental property. It is the key to solving many logarithmic problems by simplifying complex expressions. The quotient rule states that the logarithm of a quotient is equal to the difference of the logarithms.
In our example, the quotient rule is applied to show that \( \log_6\left(\frac{7}{9}\right) \) is equivalent to \( \log_6 7 - \log_6 9 \).

• Transform division into subtraction within a logarithm.
• Helps in comparing and simplifying logarithmic expressions.
• Enables easier equivalence checks by breaking down complex forms.

This rule is particularly useful when dealing with expressions that involve division inside a logarithm base. By using the quotient rule, problems that involve logarithms can become more straightforward and less daunting.