Textbook Question
Find the domain of each logarithmic function. f(x) = ln (x-2)²
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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 79
Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log(3x−3)=log(x+1)+log 4
Verified step by step guidance1
Start with the given equation: \(\log(3x - 3) = \log(x + 1) + \log 4\).
Use the logarithm property that states \(\log a + \log b = \log(ab)\) to combine the right side: \(\log(3x - 3) = \log(4(x + 1))\).
Since the logarithms on both sides have the same base and are equal, set their arguments equal: \(3x - 3 = 4(x + 1)\).
Solve the resulting linear equation for \(x\): expand the right side to get \(3x - 3 = 4x + 4\), then isolate \(x\) by moving terms appropriately.
Check the solution(s) by substituting back into the original logarithmic expressions to ensure the arguments are positive, because the domain of \(\log\) requires the input to be greater than zero.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Properties of Logarithms
Understanding the properties of logarithms, such as the product rule log(a) + log(b) = log(ab), is essential for combining or simplifying logarithmic expressions. This allows you to rewrite the equation in a more manageable form to isolate the variable.
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Change of Base Property
Domain of Logarithmic Functions
The domain of a logarithmic function includes only positive arguments because the logarithm of zero or a negative number is undefined. Identifying and restricting the domain ensures that any solutions found are valid within the original equation.
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5:26Graphs of Logarithmic Functions
Solving Logarithmic Equations
Solving logarithmic equations often involves rewriting the equation in exponential form or using properties of logarithms to isolate the variable. After simplification, you solve the resulting algebraic equation and verify solutions against the domain restrictions.
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5:02Solving Logarithmic Equations
Related Practice
Textbook Question
Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 2 log x=log 25
Textbook Question
Use a graphing utility and the change-of-base property to graph each function. y = log3 x
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Textbook Question
Use a graphing utility and the change-of-base property to graph each function. y = log2 (x + 2)
Textbook Question
Evaluate or simplify each expression without using a calculator. log 100
Textbook Question
In Exercises 74–79, solve each logarithmic equation. log4 (2x+1) = log4 (x-3) + log4 (x+5)