Domanda del libro di testo
In Exercises 105–108, evaluate each expression without using a calculator. log5 (log2 32)
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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 109
In Exercises 109–112, find the domain of each logarithmic function. f(x) = ln (x² - x − 2)
Guida verificata passo dopo passo1
Recall that the domain of a logarithmic function f(x) = ln(g(x)) requires the argument g(x) to be greater than zero, so we need to find where x² - x - 2 > 0.
Set up the inequality: .
Factor the quadratic expression: .
Determine the critical points by setting each factor equal to zero: gives , and gives .
Test intervals determined by the critical points (-∞, -1), (-1, 2), and (2, ∞) to find where the product is positive, which will give the domain of the function.
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Domain of a Function
The domain of a function is the set of all input values (x-values) for which the function is defined. For logarithmic functions, the argument inside the logarithm must be positive, so determining the domain involves finding all x-values that make the expression inside the log greater than zero.
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3:51
Domain Restrictions of Composed Functions
Properties of Logarithmic Functions
Logarithmic functions, such as the natural logarithm ln(x), are only defined for positive arguments. This means the expression inside the logarithm, here (x² - x - 2), must be greater than zero. Understanding this property is essential to correctly find the domain.
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5:26Graphs of Logarithmic Functions
Solving Quadratic Inequalities
To find where the quadratic expression (x² - x - 2) is positive, you solve the inequality x² - x - 2 > 0. This involves factoring the quadratic, finding its roots, and testing intervals to determine where the expression is positive, which helps identify the domain of the logarithmic function.
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04:03Choosing a Method to Solve Quadratics
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In Exercises 105–108, evaluate each expression without using a calculator. log2 (log3 81)
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Domanda del libro di testo
In Exercises 109–112, find the domain of each logarithmic function. f(x) = log[(x+1)/(x-5)]
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In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
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Domanda del libro di testo
In Exercises 105–108, evaluate each expression without using a calculator. log (ln e)
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Domanda del libro di testo
In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
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