Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $\ln \sqrt[5]{x}$

Verified step by step guidance

1. Recognize that the expression \( \ln 5\sqrt{x} \) can be rewritten using exponent notation. The fifth root of \( x \) is \( x^{\frac{1}{5}} \), so rewrite the expression as \( \ln \left( 5 x^{\frac{1}{5}} \right) \).
2. Use the logarithm property that \( \ln(ab) = \ln a + \ln b \) to separate the logarithm of the product into a sum: \( \ln 5 + \ln x^{\frac{1}{5}} \).
3. Apply the power rule of logarithms, which states \( \ln a^{b} = b \ln a \), to the second term: \( \ln x^{\frac{1}{5}} = \frac{1}{5} \ln x \).
4. Combine the results to express the expanded form as \( \ln 5 + \frac{1}{5} \ln x \).
5. Since \( \ln 5 \) is a constant, it can be left as is or approximated if needed, but the problem states to expand as much as possible without a calculator, so this is the fully expanded form.

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow us to expand or condense logarithmic expressions by turning multiplication into addition, division into subtraction, and exponents into coefficients. Understanding these properties is essential for simplifying and expanding logarithmic expressions.

Radicals and Exponents

Radicals like the fifth root can be expressed as fractional exponents (e.g., the fifth root of x is x^(1/5)). Converting radicals to exponents helps apply logarithm power rules effectively, making it easier to expand or simplify logarithmic expressions involving roots.

Natural Logarithm (ln)

The natural logarithm, denoted ln, is the logarithm with base e, where e is approximately 2.718. It has the same properties as other logarithms but is commonly used in calculus and algebra. Recognizing ln and its properties helps in correctly expanding and evaluating expressions without a calculator.