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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 21
Capitolo 5, Problema 21

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. logb (x2 y)

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Identify the logarithmic expression to expand: \(\log_{b} (x^{2} y)\).
Use the product property of logarithms, which states that \(\log_{b} (MN) = \log_{b} M + \log_{b} N\), to separate the logarithm of a product into a sum: \(\log_{b} (x^{2} y) = \log_{b} (x^{2}) + \log_{b} (y)\).
Apply the power property of logarithms, which states that \(\log_{b} (M^{k}) = k \log_{b} (M)\), to the term \(\log_{b} (x^{2})\): this becomes \(2 \log_{b} (x)\).
Rewrite the expanded expression combining the results: \(2 \log_{b} (x) + \log_{b} (y)\).
Check if any further simplification or evaluation is possible based on given values or context; if not, the expression is fully expanded.

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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 in simpler or expanded forms. For example, log_b(x^2 y) can be expanded using the product rule as log_b(x^2) + log_b(y).
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Change of Base Property

Product Rule of Logarithms

The product rule states that the logarithm of a product is the sum of the logarithms: log_b(MN) = log_b(M) + log_b(N). This rule helps break down complex expressions into simpler parts, making them easier to evaluate or manipulate.
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Product, Quotient, and Power Rules of Logs

Power Rule of Logarithms

The power rule states that the logarithm of a power can be expressed as the exponent times the logarithm of the base: log_b(M^k) = k * log_b(M). This is useful for moving exponents in or out of the logarithm to simplify expressions.
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Power Rules
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