Problem 41 - Intermediate Algebra

Question

Write each logarithm as a difference. Then simplify, if possible. See Example 3 . $$ \log _{6} \frac{x}{36} $$

Step-by-Step Solution

Answer

\( \log_{6} \left( \frac{x}{36} \right) = \log_{6}(x) - 2 \).

Step 1: Introduce the Logarithm Property

The expression \( \log_{b} \left( \frac{M}{N} \right) = \log_{b}(M) - \log_{b}(N) \) is a logarithmic property known as the quotient rule. It states that the log of a quotient is the difference of the logs of the numerator and denominator.

Step 2: Apply the Quotient Rule

Using the quotient rule, apply it to \( \log_{6} \left( \frac{x}{36} \right) \): \[ \log_{6} \left( \frac{x}{36} \right) = \log_{6}(x) - \log_{6}(36) \].

Step 3: Simplify the Expression

Now, simplify the expression further if possible. We know that 36 can be expressed as a power of 6: 36 = 6^2.So, \[ \log_{6}(36) = \log_{6}(6^2) = 2 \cdot \log_{6}(6) = 2. \]

Step 4: Substitute and Conclude

Replace \( \log_{6}(36) \) in the expression with 2: \[ \log_{6}(x) - \log_{6}(36) = \log_{6}(x) - 2. \] This is the simplified expression.

Key Concepts

Quotient Rule for Logarithms

The quotient rule is a fundamental concept in the study of logarithms and is incredibly handy when simplifying logarithmic expressions. It allows us to break down complex expressions involving division within a logarithmic argument into simpler terms.

• The basic formula for the quotient rule is: \( \log_{b} \left( \frac{M}{N} \right) = \log_{b}(M) - \log_{b}(N) \).
• It essentially tells us that the logarithm of a quotient is equal to the difference between the logarithm of the numerator and the logarithm of the denominator.

Understanding this rule is essential as it acts as a bridge to simplify more complicated expressions. When you see a division within a logarithm, it's a great indicator that the quotient rule can be applied, streamlining the work substantially. By using this rule, we can transform daunting expressions into more manageable ones.

Simplification of Logarithms

Once you've applied the quotient rule, the next step is simplification. Often, simplification involves recognizing parts of the logarithmic expression that can be reduced, factoring into powers, or applying known log values where possible.

• For instance, express integers like 36 in terms of powers when they relate to the logarithm base—here, \( 36 = 6^2 \).
• Apply the power rule for logarithms: \( \log_{b}(b^k) = k \cdot \log_{b}(b) \), and note that \( \log_{b}(b) = 1 \).

After applying these rules, you might find that complicated expressions collapse into far simpler forms. Simplification not only makes the expression easier to manage but also more insightful for further mathematical operations or problem-solving.

Intermediate Algebra Concepts

Logarithmic expressions and their simplifications often intersect with broader algebraic concepts, making them relevant in intermediate algebra. Understanding how these concepts work can provide deeper insight into algebraic manipulations and problem-solving.

• Exponents and logarithms are inversely related, and this relationship is pivotal when dealing with equations that include both.
• Simplification often involves recognizing numbers as powers or products of more manageable components, as seen in the conversion of 36 to \( 6^2 \).
• The interactions between different logarithmic properties, such as the quotient rule alongside the power rule, highlight the importance of understanding these underlying principles for smooth algebraic transitions.

Mastering these concepts forms a robust foundation for tackling more advanced mathematical challenges, whether you're simplifying expressions or solving logarithmic equations. Each concept builds on the previous, crafting a comprehensive toolkit for algebra.