Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $\ln \left( \frac{x^3 \sqrt{x^2 + 1}}{(x + 1)^4} \right)$

Verified step by step guidance

1. Start by writing the given logarithmic expression clearly: \(\ln\left( \frac{x^{3} \sqrt{x^{2} + 1}}{(x + 1)^{4}} \right)\).
2. Use the logarithm property for quotients: \(\ln\left( \frac{A}{B} \right) = \ln(A) - \ln(B)\), to separate the expression into two logarithms: \(\ln\left(x^{3} \sqrt{x^{2} + 1}\right) - \ln\left((x + 1)^{4}\right)\).
3. Next, apply the logarithm property for products: \(\ln(AB) = \ln(A) + \ln(B)\), to expand \(\ln\left(x^{3} \sqrt{x^{2} + 1}\right)\) into \(\ln(x^{3}) + \ln\left(\sqrt{x^{2} + 1}\right)\).
4. Rewrite the square root as an exponent: \(\sqrt{x^{2} + 1} = (x^{2} + 1)^{1/2}\), and use the power rule for logarithms: \(\ln\left(A^{r}\right) = r \ln(A)\), to express \(\ln\left(\sqrt{x^{2} + 1}\right)\) as \(\frac{1}{2} \ln(x^{2} + 1)\).
5. Similarly, apply the power rule to \(\ln(x^{3})\) and \(\ln\left((x + 1)^{4}\right)\) to get \$3 \ln(x)\( and \)4 \ln(x + 1)$ respectively. Combine all parts to write the fully expanded expression.

Key Concepts

Properties of Logarithms

Properties of logarithms include the product, quotient, and power rules, which allow the expansion or simplification of logarithmic expressions. For example, ln(ab) = ln(a) + ln(b), ln(a/b) = ln(a) - ln(b), and ln(a^n) = n ln(a). These rules help break down complex expressions into simpler sums and differences of logarithms.

Simplifying Radicals and Exponents

Simplifying expressions involving radicals and exponents is essential before applying logarithmic properties. For instance, the square root √(x² + 1) can be left as is, but recognizing powers like x³ or (x + 1)^4 helps in applying the power rule of logarithms effectively.

Domain Considerations for Logarithmic Functions

The domain of a logarithmic function requires the argument to be positive. When expanding ln[(x³(√(x² + 1)))/(x + 1)⁴], it is important to consider values of x that keep the entire expression inside the logarithm positive to ensure the expression is defined.