Problem 274 - Calculus Volume 1

Question

Use properties of logarithms to write the expressions as a sum, difference, and/or product of logarithms. \(\log _{4} \frac{\sqrt[3]{x y}}{64}\)

Step-by-Step Solution

Answer

\( \frac{1}{3}(\log_{4}x + \log_{4}y) - 3 \)

Step 1: Apply the Quotient Rule of Logarithms

Start by using the quotient rule of logarithms that states \( \log_b \frac{M}{N} = \log_b M - \log_b N \). Here, \( M \) is \( \sqrt[3]{xy} \) and \( N \) is \( 64 \). Thus, we can rewrite the expression as: \[ \log_{4} \sqrt[3]{xy} - \log_{4} 64 \]

Step 2: Apply the Exponent Rule to the Radicand

The cube root \( \sqrt[3]{xy} \) can be expressed as \( (xy)^{\frac{1}{3}} \). Use the power rule \( \log_b M^n = n \cdot \log_b M \) to rewrite the term: \[ \frac{1}{3} \cdot \log_{4} (xy) \] So, the first part of the expression becomes: \[ \frac{1}{3} \cdot ( \log_{4} x + \log_{4} y) \] by applying the product rule \( \log_b (MN) = \log_b M + \log_b N \).

Step 3: Simplify the Logarithm of 64

Recognize that \( 64 = 4^3 \), so \( \log_{4} 64 = 3 \cdot \log_{4} 4 = 3 \) because \( \log_{4} 4 = 1 \).

Step 4: Combine and Simplify all Terms

Combine all the terms from the previous steps to get the final expression: \[ \frac{1}{3} ( \log_{4} x + \log_{4} y ) - 3 \]

Key Concepts

Quotient Rule of Logarithms

The quotient rule of logarithms is a powerful property that simplifies division expressions into a difference of logs. It states that the logarithm of a quotient is equal to the difference between the logs of the numerator and the denominator. Mathematically, it is expressed as:

• \( \log_b \frac{M}{N} = \log_b M - \log_b N \)

In our problem, the original expression to simplify is \( \log_{4} \frac{\sqrt[3]{xy}}{64} \). Applying the quotient rule allows us to separate this into two simpler terms: \( \log_{4} \sqrt[3]{xy} \) and \( \log_{4} 64 \). This transformation is often the first step when dealing with complex logarithmic expressions, as it breaks down a fraction into easily manageable parts.

Product Rule of Logarithms

The product rule is another essential tool when simplifying logarithmic expressions. It enables you to simplify expressions involving multiplication within a log by transforming them into a sum. The product rule is expressed as:

• \( \log_b (MN) = \log_b M + \log_b N \)

In this problem, after using the quotient rule, we applied the product rule to the expression \( \log_{4} (xy) \). Based on the product rule, \( \log_{4} (xy) \) can be rewritten as \( \log_{4} x + \log_{4} y \).
This decomposition is particularly useful when variables are involved in the product, as it allows each component to be treated separately.
Breaking it down into separate logs makes the expression easier to handle and simplifies the problem further.

Power Rule of Logarithms

The power rule of logarithms helps you handle expressions with exponents within the log. It simplifies these by moving the exponent in front of the log, making it a coefficient. The power rule is given by:

• \( \log_b M^n = n \cdot \log_b M \)

In our exercise, the expression \( \sqrt[3]{xy} \) is understood as \( (xy)^{\frac{1}{3}} \). When applying the power rule here, the exponent \( \frac{1}{3} \) is brought in front, allowing the rewritten form \( \frac{1}{3} \cdot \log_{4} (xy) \). Subsequently, we expand \( \log_{4} (xy) \) using the product rule resulting in:

• \( \frac{1}{3} (\log_{4} x + \log_{4} y) \)

The power rule simplifies logs with exponents and is pivotal in handling roots through their fractional exponents.

Logarithmic Expressions Simplification

Simplification of logarithmic expressions involves applying logarithm rules to transform a complex log equation into its simplest form. This process not only simplifies calculations but also aids in solving equations and understanding the inherent relationships.

• Recognize patterns: Identify parts of the logarithmic expression which fit the quotient, product, or power rules.
• Apply rules step-by-step: Carefully use each applicable rule one after another, ensuring correct transformations.
• Reorganization: Combine like terms or constants efficiently.

In our exercise, combining all element applications resulted in \( \frac{1}{3} (\log_{4} x + \log_{4} y) - 3 \). Recognizing constants, with \( \log_{4} 64 \) equating to 3, further simplified the problem.
Mastering simplification through these rules enables tackling more complex logarithmic equations with ease.