Problem 32 - 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. $$\log \sqrt[5]{\frac{x}{y}}$$

Step-by-Step Solution

Answer

The expanded form of the given expression \( \log \sqrt[5]{\frac{x}{y}} \) is \( \frac{1}{5} \log x - \frac{1}{5} \log y \).

Step 1: Applying property of root inside logarithm

The property of logarithm states that the root within the logarithm can be taken out as a fractional multiplier of the log. Therefore, \( \log \sqrt[5]{\frac{x}{y}} \) can be written as \( \frac{1}{5} \log \frac{x}{y} \).

Step 2: Applying property of quotient inside logarithm

Another property of logarithms is that the log of a quotient is the difference of the logs. This property allows us to split the logarithm of the fraction into difference of two separate logs. Therefore, \( \frac{1}{5}\log\frac{x}{y} \) can be written as \( \frac{1}{5} (\log x - \log y) \).

Step 3: Distributing the Fraction

Now we distribute the \( \frac{1}{5} \) to both terms within the parentheses to obtain \( \frac{1}{5} \log x - \frac{1}{5} \log y \). This is the expanded form of the original logarithmic expression.