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
In Exercises 74–79, solve each logarithmic equation. log4 (2x+1) = log4 (x-3) + log4 (x+5)
Verified step by step guidance1
Recall the logarithmic property that allows you to combine the sum of logarithms with the same base: \(\log_b A + \log_b B = \log_b (A \times B)\). Apply this to the right side of the equation to combine the two logarithms into one.
Rewrite the equation using the property: \(\log_4 (2x + 1) = \log_4 ((x - 3)(x + 5))\).
Since the logarithms on both sides have the same base and are equal, set their arguments equal to each other: \(2x + 1 = (x - 3)(x + 5)\).
Expand the right side by multiplying the binomials: \((x - 3)(x + 5) = x^2 + 5x - 3x - 15 = x^2 + 2x - 15\).
Set up the equation \(2x + 1 = x^2 + 2x - 15\) and then rearrange all terms to one side to form a quadratic equation: \(0 = x^2 + 2x - 15 - 2x - 1\), which simplifies to \(0 = x^2 - 16\). From here, solve for \(x\).
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4mKey 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, quotient, and power rules, is essential. In this problem, the product rule allows combining the sum of logarithms on one side into a single logarithm, simplifying the equation for easier solving.
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Change of Base Property
Solving Logarithmic Equations
Solving logarithmic equations often involves rewriting the equation using logarithm properties and then converting it to an exponential form or equating the arguments of the logarithms when the bases are the same. This approach helps isolate the variable and find its value.
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5:02Solving Logarithmic Equations
Domain Restrictions of Logarithmic Functions
Logarithmic functions are only defined for positive arguments. When solving equations, it is crucial to check that the solutions satisfy the domain restrictions (arguments inside the logs must be greater than zero) to ensure valid answers.
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3:51Domain Restrictions of Composed Functions
Related Practice
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
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
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Textbook Question
Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. logπ 63