In Exercises 50–53, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $\ln \sqrt[3]{\frac{x}{e}}$

Verified step by step guidance

1. Recognize that the expression involves the natural logarithm of a cube root: \(\ln \sqrt[3]{\frac{x}{e}}\).
2. Rewrite the cube root as an exponent: \(\ln \left( \frac{x}{e} \right)^{\frac{1}{3}}\).
3. Use the logarithm power rule to bring the exponent in front: \(\frac{1}{3} \ln \left( \frac{x}{e} \right)\).
4. Apply the logarithm quotient rule to separate the fraction inside the logarithm: \(\frac{1}{3} \left( \ln x - \ln e \right)\).
5. Recall that \(\ln e = 1\), so simplify the expression to \(\frac{1}{3} (\ln x - 1)\).

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow us to rewrite logarithmic expressions by expanding or condensing them. For example, ln(a/b) = ln(a) - ln(b) and ln(a^r) = r ln(a). These properties are essential for simplifying and expanding logarithmic expressions.

Natural Logarithm (ln)

The natural logarithm, denoted ln, is the logarithm with base e, where e ≈ 2.718. It is the inverse function of the exponential function e^x. Understanding ln is crucial for manipulating expressions involving e and for applying logarithmic properties correctly.

Radicals and Exponents

Radicals such as cube roots can be expressed as fractional exponents, e.g., ∛x = x^(1/3). Converting radicals to exponents helps apply logarithmic power rules effectively. This conversion simplifies the expansion of logarithmic expressions involving roots.