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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 53
Capitolo 5, Problema 53

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. 2 logb x + 3 logb y

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Identify the given expression: \(2 \log_{b} x + 3 \log_{b} y\).
Recall the logarithmic property that allows you to move coefficients as exponents inside the logarithm: \(a \log_{b} M = \log_{b} (M^{a})\).
Apply this property to each term: \(2 \log_{b} x = \log_{b} (x^{2})\) and \(3 \log_{b} y = \log_{b} (y^{3})\).
Use the logarithmic property for addition: \(\log_{b} A + \log_{b} B = \log_{b} (A \times B)\) to combine the two terms into a single logarithm.
Write the final condensed expression as \(\log_{b} (x^{2} y^{3})\), which is a single logarithm with coefficient 1.

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Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow combining multiple logarithmic terms into a single logarithm by converting coefficients into exponents and combining sums or differences into products or quotients.
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Change of Base Property

Power Rule of Logarithms

The power rule states that a coefficient in front of a logarithm can be rewritten as an exponent inside the logarithm, i.e., a·log_b(x) = log_b(x^a). This is essential for condensing expressions with coefficients into a single logarithm.
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Power Rules

Condensing Logarithmic Expressions

Condensing logarithmic expressions means rewriting a sum or difference of logarithms as a single logarithm. This involves applying the product or quotient rules after using the power rule to handle coefficients, simplifying the expression into one logarithm with coefficient 1.
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Expand & Condense Log Expressions
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