Problem 62 - Elementary and Intermediate Algebra

Question

Write logarithm as the sum and/or difference of logarithms of a single quantity. Then simplify, if possible. \(\log _{3} \frac{\sqrt[4]{x}}{y z}\)

Step-by-Step Solution

Answer

\( \frac{1}{4} \log_3(x) - \log_3(y) - \log_3(z) \)

Step 1: Write the Fraction as a Difference

Use the property of logarithms that states: \( \log_b \left( \frac{a}{c} \right) = \log_b(a) - \log_b(c) \). Apply this to the given expression \( \log_3 \frac{\sqrt[4]{x}}{yz} \). This can be written as: \[ \log_3 \left( \sqrt[4]{x} \right) - \log_3(yz) \]

Step 2: Apply Logarithm of a Power Rule

Use the logarithm rule \( \log_b(a^n) = n \cdot \log_b(a) \). Rewrite \( \log_3 \left( \sqrt[4]{x} \right) \) as this is \( x^{1/4} \). So, \[ \log_3(x^{1/4}) = \frac{1}{4} \cdot \log_3(x) \] Combine this into the expression from Step 1: \[ \frac{1}{4} \log_3(x) - \log_3(yz) \]

Step 3: Apply Product to Sum Rule

Use the property \( \log_b(ac) = \log_b(a) + \log_b(c) \). Apply this to the term \( \log_3(yz) \): \[ \log_3(yz) = \log_3(y) + \log_3(z) \] Substitute this back into the expression from Step 2: \[ \frac{1}{4} \log_3(x) - (\log_3(y) + \log_3(z)) \]

Step 4: Distribute the Negative Sign

Apply the negative sign to the terms within the parentheses: \[ \frac{1}{4} \log_3(x) - \log_3(y) - \log_3(z) \] This is the expression for the logarithm written as the sum or difference of logarithms.

Key Concepts

Logarithmic Properties

Understanding the properties of logarithms is crucial for rewriting complex logarithmic expressions in simpler, more workable forms. These properties allow us to break down logs into sums or differences, making them easier to handle. Let's explore some key properties that are often used:

• Quotient Rule: This allows us to express a log of a fraction as the difference of two logs: \( \log_b \left( \frac{a}{c} \right) = \log_b(a) - \log_b(c) \). This property is handy when dealing with ratios.
• Product Rule: Here, a log of a product becomes a sum: \( \log_b(ac) = \log_b(a) + \log_b(c) \). This makes multiplication inside a logarithm easier to manage.
• Power Rule: It helps simplify logs involving exponents. Express \( \log_b(a^n) \) as \( n \cdot \log_b(a) \). This is useful when you're dealing with powers and roots.

By applying these properties, you can transform a complex log into a combination of simpler ones, enabling easier calculations or further algebraic manipulations.

Simplification of Expressions

When faced with complicated expressions, simplification makes them more manageable and easier to work with. Here's a brief guide on simplifying expressions involving logs:Start by applying the Quotient Rule, which allows breaking down a log of a fraction into a difference of two logs. In the example \( \log_3 \frac{\sqrt[4]{x}}{yz} \), this step simplifies the fraction inside the logarithm:\[ \log_3 \left( \sqrt[4]{x} \right) - \log_3(yz) \]Next, use the Power Rule for expressions involving exponents, transforming them into simpler terms. The term \( \log_3 \left( \sqrt[4]{x} \right) \) becomes:\[ \frac{1}{4} \cdot \log_3(x) \]Finally, apply the Product Rule to break a product inside a log into a sum of separate logs. This transforms \( \log_3(yz) \) into:\[ \log_3(y) + \log_3(z) \]Each step reduces the complexity, resulting in an expression that is not only simpler but also more intuitive for further solving or analysis.

Logarithmic Identities

Logarithmic identities are standard equations that hold true for all valid inputs, helping to relate basic logarithmic operations with one another. When simplifying expressions or solving equations involving logarithms, these identities come in handy.Some core logarithmic identities include:

• Logarithm of 1: \( \log_b(1) = 0 \) for any base \( b \).
• Logarithm of the Base: \( \log_b(b) = 1 \).
• Change of Base Formula: \( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \) allows switching between different logarithmic bases.

These identities serve as basic principles to consolidate expressions and verify results.In practice, these identities might not be directly visible in the simplification process, but they underlie most transformations and simplifications in logarithmic problem-solving. Understanding how and when to apply these identities allows for more efficient and accurate manipulations of logarithmic expressions.