Textbook Question
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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Table of contents
- 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
Chapter 5, Problem 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)
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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Related Practice
Textbook Question
The graph of an exponential function is given. Select the function for each graph from the following options: f(x) = 4x, g(x) = 4-x, h(x) = -4-x, r(x) = -4-x+3
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Textbook Question
Write each equation in its equivalent exponential form. 2 = log9 x
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Textbook Question
Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 5x=125
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Textbook Question
The graph of an exponential function is given. Select the function for each graph from the following options: f(x) = 4x, g(x) = 4-x, h(x) = -4-x, r(x) = -4-x+3
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Textbook Question
Write each equation in its equivalent exponential form. 2 = log3 x
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