Write the quantity using sums and differences of simpler...

Write the quantity using sums and differences of simpler logarithmic expressions. Express the answer so that logarithms of products, quotients, and powers do not appear.
(a) $\log _{b} \sqrt[3]{\frac{(x-1)^{2}(x-2)}{(x+2)^{2}(x+1)}}$
(b) $\ln \left(\frac{e-1}{e+1}\right)^{3 / 2}$

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Key Concepts

Properties of Logarithms

This concept covers the fundamental rules such as the product rule (log_b(MN) = log_b M + log_b N), the quotient rule (log_b(M/N) = log_b M - log_b N), and the power rule (log_b(M^p) = p · log_b M). Understanding these properties is essential for decomposing complex logarithmic expressions into simpler sums and differences.

Representation of Roots as Exponents

Any radical expression can be rewritten using fractional exponents (for example, the nth root of M is expressed as M^(1/n)). This conversion is critical because it allows the power rule of logarithms to be applied, thereby simplifying expressions that involve roots before breaking them down into simpler logarithmic terms.

Decomposition of Complex Logarithmic Expressions

Breaking down a complex logarithmic expression into smaller, more manageable parts involves systematically applying logarithmic rules to eliminate products, quotients, and exponents inside the logarithm. This step-by-step simplification helps in analyzing and understanding the overall structure of the expression.

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