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In Exercises 1–10, approximate each number using a calculator. Round your answer to three decimal places. 3√5
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Ch. 4 - Exponential and Logarithmic Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 5, Problema 3
Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log7 (7x)
Guida verificata passo dopo passo1
Identify the logarithmic expression given: \(\log_{7}(7x)\).
Recall the logarithmic property that states \(\log_b(mn) = \log_b(m) + \log_b(n)\), which allows us to expand the log of a product into a sum of logs.
Apply this property to the expression: \(\log_{7}(7x) = \log_{7}(7) + \log_{7}(x)\).
Evaluate \(\log_{7}(7)\) using the fact that \(\log_b(b) = 1\) for any base \(b\), so \(\log_{7}(7) = 1\).
Write the fully expanded form as \(1 + \log_{7}(x)\), which is the simplified expanded expression.
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Properties of Logarithms
Properties of logarithms include rules such as the product, quotient, and power rules that allow the expansion or simplification of logarithmic expressions. For example, log_b(MN) = log_b(M) + log_b(N) helps break down complex expressions into simpler parts.
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Change of Base Property
Logarithm of a Base Raised to a Power
The logarithm of a base raised to the same base simplifies to the exponent, i.e., log_b(b) = 1. This property is useful for evaluating expressions like log_7(7), which equals 1, simplifying the overall expression.
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Evaluating Logarithmic Expressions Without a Calculator
Some logarithmic expressions can be evaluated exactly by recognizing patterns or using properties, such as when the argument is a power of the base. This avoids approximation and provides exact values, enhancing understanding of logarithmic behavior.
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5:14Evaluate Logarithms
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