Problem 64 - Algebra and Trigonometry

Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \frac{1}{3}\left(\log _{4} x-\log _{4} y\right) $$

Step-by-Step Solution

Answer

The condensed form of the given logarithmic expression is \(\log_4 \sqrt[3]{\frac{x}{y}}\).

Step 1: Apply Logarithmic Subtraction Rule

For a given base, the subtraction between two logarithms equates to the logarithm of the division of their input. Now, apply the logarithmic subtraction rule to the equation: \(\frac{1}{3}(\log_4 x- \log_4 y) = \frac{1}{3} \log_4 \frac{x}{y}\)

Step 2: Apply Logarithmic Power Rule to Simplify

The Power Rule states that a logarithm raised to a power is the same as the logarithm of the input raised to that power. Apply it to the equation: \(\frac{1}{3} \log_4 \frac{x}{y} = \log_4 \sqrt[3]{\frac{x}{y}}\)

Key Concepts

Logarithmic Subtraction Rule

Understanding the logarithmic subtraction rule can simplify complex expressions significantly. In essence, this rule allows us to combine two logarithms with the same base and a subtraction sign between them into a single logarithm.

When you see something like \( \log_b{x} - \log_b{y} \), the rule tells us that we can rewrite this as one logarithm: \( \log_b{\frac{x}{y}} \). This transformation is handy because it simplifies the expression and reduces the number of logarithms we're working with.

For example, if you start with \( \frac{1}{3}(\log_4 x - \log_4 y) \), you apply this rule to get \( \frac{1}{3} \log_4 \frac{x}{y} \). This technique is not only simple but also crucial for condensing logarithmic expressions and further simplification.

Logarithmic Power Rule

The logarithmic power rule is a powerful tool when dealing with exponents within logarithmic functions. It states that the exponent of a logarithmic argument can be moved in front of the logarithm as a multiplier. In notation, this reads as \( \log_b(x^p) = p \cdot \log_b(x) \).

Applying this rule can transform expressions in ways that make them more manageable. For our earlier example, the expression \( \frac{1}{3} \log_4 \frac{x}{y} \) can be rewritten using the power rule as \( \log_4 (\frac{x}{y})^{\frac{1}{3}} \), which is the logarithm of the cube root of \( \frac{x}{y} \). Understanding and applying this rule makes it possible to manipulate logarithmic expressions into simpler forms or forms that are better suited for further calculations.

Simplifying Logarithmic Expressions

When faced with logarithmic expressions, our goal is often to simplify them to the most basic form possible. Simplification involves applying the rules of logarithms, like the subtraction and power rules, to break down the expressions into single logarithms if feasible.

Simplification not only makes the expressions easier to understand but also prepares them for evaluation or further algebraic manipulation. For instance, reducing an expression to a single logarithm might reveal that it's possible to solve for an unknown, or to evaluate the expression without a calculator. By thinking of logarithmic simplification as a systematic process, you're less likely to be overwhelmed by complex expressions and more likely to identify the steps needed to transform them.

Condensing Logarithms

Condensing logarithms is the reverse process of expanding them - it's essentially about combining multiple logarithmic terms into a single term. This comes in handy in various mathematical and real-world applications where we need to express complex relationships in a more digestible format.

The process often involves using the logarithmic subtraction and power rules, as well as other properties like the product rule. In practice, condensing is about seeing the bigger picture; it's about transforming an untidy bunch of logs into one neatly packaged mathematical expression.

The exercise provided is a good example, where \( \frac{1}{3}(\log_4 x - \log_4 y) \) is condensed to \( \log_4 \sqrt[3]{\frac{x}{y}} \), a single log expression. Mastery of this skill makes navigating through logarithms much less intimidating.