In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. ln[(x^3(√(x^2 + 1))/(x + 1)^4]

Verified step by step guidance

1. <Apply the logarithm of a quotient rule: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \).>
2. <Apply the logarithm of a product rule: \( \ln(ab) = \ln(a) + \ln(b) \).>
3. <Apply the power rule for logarithms: \( \ln(a^b) = b \cdot \ln(a) \).>
4. <Simplify each term separately using the rules above.>
5. <Combine all simplified terms to express the expanded form of the logarithmic expression.>

Key Concepts

Properties of Logarithms

The properties of logarithms include rules that simplify the manipulation of logarithmic expressions. Key properties include the product rule (log(a*b) = log(a) + log(b)), the quotient rule (log(a/b) = log(a) - log(b)), and the power rule (log(a^b) = b*log(a)). Understanding these properties is essential for expanding logarithmic expressions effectively.

Natural Logarithm (ln)

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It is commonly used in calculus and higher mathematics. Recognizing that ln has specific properties similar to other logarithms helps in applying the logarithmic rules correctly when expanding expressions.

Radicals and Exponents

Radicals, such as square roots, can be expressed as exponents (e.g., √(x) = x^(1/2)). This understanding is crucial when dealing with logarithmic expressions that involve roots, as it allows for the application of the power rule in logarithms. Converting radicals to exponent form simplifies the expansion process.