Problem 20 - College Algebra Essentials

Question

Use properties of logarithms to expand 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 logarithmic expression is (1/7)\( \ln x\)

Step 1: Identify the root in logarithmic expression

Recognizing the root in the logarithmic expression, realize that \(\sqrt[7]{x}\) stands for seventh root of \(x\). This will be altered using logarithm properties.

Step 2: Apply the property of logarithms

According to the properties of logarithms, the logarithm of a root is the exponent as a fraction times the logarithm. Hence, we rewrite the expression following this property, \(\ln \sqrt[7]{x}\) = (1/7)\( \ln x\)

Step 3: Simplify and Evaluate

The given expression has been simplified to (1/7)\( \ln x\). This can't be evaluated any further without specific value to \( x\)

Key Concepts

Understanding Logarithmic Expressions

To start off, logarithmic expressions are a way to rewrite exponential equations in an alternative form. The expression \( \ln \sqrt[7]{x} \) involves a logarithm with a radical (meaning root). A radical can be denoted by an exponent fraction, for instance, a seventh root can be expressed as a power of \(1/7\).

When working through problems with logarithmic expressions, recognize that the logarithm is asking, \