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

Step-by-Step Solution

Answer

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

Step 1: Recognize the Expression as a Logarithmic Function

First, acknowledge that the presented problem is a logarithm. In this instance, a natural logarithm is denoted by \(ln\).

Step 2: Convert the Root to an Exponent

\(\sqrt[5]{x}\) can be rewritten as \(x^{1/5}\) since the fifth root of x is equivalent to \(x\) raised to the power \(1/5\). So, \( \ln \sqrt[5]{x} \) simplifies to \( \ln x^{1/5}\).

Step 3: Apply Logarithm Properties

By the power rule for logarithms, the exponent in the argument of the logarithm can be brought out as a coefficient. Therefore, \( \ln x^{1/5} \) transforms to \( \frac{1}{5} \ln x \).