Problem 31 - College Algebra

Question

In Exercises \(1-40,\) use properties of logarithms to expand each 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 \( \log \sqrt[3]{\frac{x}{y}} \) is \( \frac{1}{3} \log (x) - \frac{1}{3} \log (y) \).

Step 1: Break down the cube root using exponentiation rules

In this step, we break down the cube root using exponentiation rules, such that \(\sqrt[3]{\frac{x}{y}}\) is equivalent to \((\frac{x}{y})^{1/3}\). The logarithmic expression now becomes \( \log \left(\frac{x}{y} \right)^{1/3}\).

Step 2: Apply the logarithmic power rule

The logarithmic power rule states that \( \log_b(m^n) = n \log_b(m) \). Applying this rule, we get: \( \frac{1}{3} \log \left(\frac{x}{y} \right) \).

Step 3: Apply the quotient property of logarithms

The quotient property tells us that \( \log_b \left(\frac{m}{n} \right) = \log_b(m) - \log_b(n) \). Using this property, our expression simplifies down to: \( \frac{1}{3} (\log (x) - \log (y)) \).

Step 4: Distribute the 1/3 factor

Apply the 1/3 factor to both terms in the brackets to get the final answer: \( \frac{1}{3} \log (x) - \frac{1}{3} \log (y) \).