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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblema 35
Capitolo 5, Problema 35

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log5x2y243\(\log\)_5 \(\sqrt\)[3]{\(\frac{x^2 y}{24}\)}

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Identify the logarithmic expression to expand: \(\log_{5} \sqrt[3]{\frac{x^{2} y}{24}}\).
Rewrite the cube root as a fractional exponent: \(\log_{5} \left( \frac{x^{2} y}{24} \right)^{\frac{1}{3}}\).
Use the power rule of logarithms to bring the exponent in front: \(\frac{1}{3} \log_{5} \left( \frac{x^{2} y}{24} \right)\).
Apply the quotient rule of logarithms to separate numerator and denominator: \(\frac{1}{3} \left( \log_{5} (x^{2} y) - \log_{5} 24 \right)\).
Use the product rule of logarithms to expand the numerator: \(\frac{1}{3} \left( \log_{5} x^{2} + \log_{5} y - \log_{5} 24 \right)\), then apply the power rule to \(\log_{5} x^{2}\) as \(2 \log_{5} x\).

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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, log_b(MN) = log_b(M) + log_b(N), log_b(M/N) = log_b(M) - log_b(N), and log_b(M^p) = p·log_b(M).
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Radicals and Exponents

Understanding how to express radicals as fractional exponents is essential. For instance, the cube root of a quantity can be written as that quantity raised to the 1/3 power. This conversion helps apply the power rule of logarithms effectively.
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Rational Exponents

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves breaking down complex arguments into simpler parts using the properties of logarithms. This process often includes factoring, separating products and quotients, and applying exponents to isolate terms for easier evaluation.
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Simplifying Algebraic Expressions
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