Problem 19 - 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. $$ \ln \sqrt[5]{x} $$

Step-by-Step Solution

Answer

The expanded form of \( \ln \sqrt[5]{x} \) is \( \frac{1}{5}\ln x \).

Step 1: Identify the property of logarithm

Look at the logarithmic expression and identify the properties of logarithms applicable here. The expression is a natural logarithm of a fifth root of some number x. The specific logarithmic property we will use here is that the logarithm of a root is the same as dividing the logarithm of the number by the root. Therefore, \( \ln \sqrt[5]{x} = \frac{1}{5}\ln x \).

Step 2: Apply the Logarithmic Property

Next step is applying the logarithmic property on \( \ln \sqrt[5]{x} \). Rewrite \( \sqrt[5]{x} \) as \( x^{1/5} \). Now applying the logarithmic property, coefficient of \( x^{1/5} \) becomes the multiplier to \( \ln x \). So, the expanded form of \( \ln \sqrt[5]{x} = \frac{1}{5}\ln x \).