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) $\log _{b} \sqrt[3]{\frac{x+3}{x}}$
(b) $\ln \frac{1}{\sqrt{x^{2}+x+1}}$

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Key Concepts

Power Rule for Logarithms

The power rule states that the logarithm of a quantity raised to an exponent is equal to the exponent multiplied by the logarithm of the base quantity. In mathematical terms, log_b(a^r) = r · log_b(a). This rule is pivotal when separating coefficients from the logarithm, particularly after converting roots into fractional exponents.

Radical Expressions as Fractional Exponents

This concept involves rewriting radical expressions as expressions with fractional exponents. For example, the nth root of any quantity can be expressed as the quantity raised to the 1/n power. This conversion is essential when applying logarithmic rules because it allows one to handle roots with the power rule of logarithms, simplifying the expression into a multiple of a logarithm.

Quotient Rule for Logarithms

The quotient rule for logarithms explains that the logarithm of a quotient is equal to the difference between the logarithms of the numerator and the denominator. Formally, log_b(a/c) = log_b(a) - log_b(c). This rule facilitates breaking down a logarithm of a fraction into simpler, separate logarithmic expressions, which is useful in simplifying complex logarithmic expressions.

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