Problem 31 - Introductory and Intermediate Algebra for College Students 4th

Question

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 expansion of \( \log \sqrt[3]{\frac{x}{y}} \) is \( \frac{1}{3} \left( \log x - \log y \right) \).

Step 1: Convert the Cubic Root into a Fractional Power

To manipulate the expression easier, it's beneficial to first convert the cubic root into a fractional power. Doing so, we get \( \log \left(\frac{x}{y}\right)^\frac{1}{3} \).

Step 2: Use the Power Rule of Logarithms

Using the power rule of logarithms, the power in the argument can be brought down as a coefficient. So, the expression becomes \( \frac{1}{3}\log \left(\frac{x}{y}\right) \).

Step 3: Apply the Quotient Rule of Logarithms

By the quotient rule for logarithms, the log of a quotient is equal to the difference of logs. Therefore, we can separate the fraction: \( \frac{1}{3} \left( \log x - \log y \right) \). Thus, we have expanded the given expression and evaluated as much as possible without using the calculator.