Problem 20 - 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. $$\ln \sqrt[7]{x}$$

Step-by-Step Solution

Answer

The expanded form of the expression \( \ln \sqrt[7]{x} \) is \( \frac{1}{7} \ln{x} \).

Step 1: Convert the root into power

Firstly, convert the seventh root of \( x \) into an equivalent expression using exponent form. The seventh root of \( x \) can be rewritten as \( x^{1/7} \). So, the expression becomes \( \ln{x^{1/7}} \).

Step 2: Apply logarithmic property

Apply the logarithmic property \( \ln(a^b) = b \ln(a) \) on the converted expression. This leads to the expansion of \( \ln{x^{1/7}} \) as \( 1/7 \cdot \ln{x} \).

Step 3: Final expression

The final expression after expansion using properties of logarithms becomes \( 1/7 \cdot \ln{x} \).