Problem 31 - College Algebra Essentials

Question

Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\log \sqrt[3]{\frac{x}{y}}\)

Step-by-Step Solution

Answer

The expanded form of the given logarithmic expression is \( \frac{1}{3}*\log(x) - \frac{1}{3}*\log(y) \)

Step 1: Change the Logarithm of a Root into a Division

The cube root \(\sqrt[3]{\frac{x}{y}}\) can be re-written as \( (\frac{x}{y})^{1/3} \). This helps in converting the root into a simple power, which can be further simplified using the rule \( log_b(m^n) = n * log_b(m) \). The expression becomes \( \frac{1}{3}*\log( \frac{x}{y}) \).

Step 2: Translate the Logarithm of a Division into a Subtraction

The fraction within the brackets can be translated into a subtraction using the logarithmic rule \( log_b(m/n) = log_b(m) - log_b(n) \). This divide can be turned into a subtraction: \( \frac{1}{3} [\log(x) - \log(y) ] \).

Step 3: Distribute the Fraction

Distribute the fraction into the bracket to simplify it further: \( \frac{1}{3}*\log(x) - \frac{1}{3}*\log(y) \).

Key Concepts

Properties of Logarithms

Logarithms have unique properties that make complex expressions simpler to work with. Understanding these properties is key to manipulating and evaluating logarithmic expressions without a calculator. Some of the most essential properties include:

• Product Property: This tells us that the logarithm of a product is equal to the sum of the logarithms. Mathematically, this is expressed as \( \log_b(m \, n) = \log_b(m) + \log_b(n) \).
• Quotient Property: This indicates that the logarithm of a division is the logarithm of the numerator minus the logarithm of the denominator, \( \log_b(m/n) = \log_b(m) - \log_b(n) \).
• Power Property: This states that the logarithm of a power can be simplified by multiplying the exponent by the logarithm of the base, \( \log_b(m^n) = n \cdot \log_b(m) \).

By applying these properties, we can expand, simplify, and evaluate logarithmic expressions efficiently.

Expanding Logarithms

Expanding logarithms involves using the properties of logarithms to break down more complex logarithmic expressions into simpler components. This process is not only helpful for simplifying but also for solving logarithmic equations.

Example of Expanding Logarithms

Consider the expression \( \log \sqrt[3]{x/y} \). Initially, it's a cube root, which can be rewritten using the power property: \((x/y)^{1/3}\). By changing the logarithm of a root into a division, we rewrite it as \( \frac{1}{3} \cdot \log{(x/y)} \).

Using the Quotient Property

Next, we apply the quotient property \( \log(m/n) \) becomes \( \log(m) - \log(n) \), thus we have: \( \frac{1}{3} [ \log(x) - \log(y) ] \).Breaking down complex expressions into their fundamental components is crucial for understanding and solving more challenging logarithmic problems.

Logarithmic Rules

Logarithmic rules are practical for converting, expanding, and simplifying expressions. These rules are based on the core properties of logarithms, providing a structured approach to handling expressions.

Applying Logarithmic Rules

Let's look again at our expression \( \log \sqrt[3]{x/y} \). We began by rewriting our expression, using the power rule (\( \log(m^n) = n \cdot \log(m) \)), to \( \frac{1}{3} \cdot \log(x/y) \). Applying the power rule helps us separate the fraction involving the logarithms.

Consistent Use of the Quotient Rule

We then use the quotient rule, which transforms our expression into two separate logarithms, showing subtraction: \( \log(x) - \log(y) \). Distributing the \( \frac{1}{3} \) across these helps us finally reach the expanded form: \( \frac{1}{3} \cdot \log(x) - \frac{1}{3} \cdot \log(y) \).Understanding these rules allows us to handle logarithms with more confidence and facilitates our ability to solve logarithmic equations efficiently.