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.
logb (xy)5 = (logb x + logb y)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 111
In Exercises 109–112, find the domain of each logarithmic function. f(x) = log[(x+1)/(x-5)]
Guida verificata passo dopo passo1
Recall that the domain of a logarithmic function \( f(x) = \log(g(x)) \) consists of all values of \( x \) for which the argument \( g(x) \) is positive, i.e., \( g(x) > 0 \).
Identify the argument of the logarithm in the function: \( \frac{x+1}{x-5} \). We need to find where \( \frac{x+1}{x-5} > 0 \).
Determine the critical points where the numerator or denominator is zero: \( x+1=0 \Rightarrow x=-1 \) and \( x-5=0 \Rightarrow x=5 \). These points divide the number line into intervals to test.
Test the sign of \( \frac{x+1}{x-5} \) in each interval: \( (-\infty, -1) \), \( (-1, 5) \), and \( (5, \infty) \). For each interval, pick a test value and check if the fraction is positive.
Combine the intervals where the fraction is positive, excluding points where the denominator is zero (since division by zero is undefined), to write the domain of \( f(x) \).
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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 log must be positive, so determining the domain involves finding all x-values that make the argument greater than zero.
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3:51
Domain Restrictions of Composed Functions
Properties of Logarithmic Functions
A logarithmic function log_b(A) is defined only when the argument A is positive (A > 0). This means that for f(x) = log[(x+1)/(x-5)], the expression (x+1)/(x-5) must be greater than zero to ensure the function is valid.
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5:26Graphs of Logarithmic Functions
Inequalities Involving Rational Expressions
To find where (x+1)/(x-5) > 0, one must solve a rational inequality. This involves determining the sign of the numerator and denominator, identifying critical points where the expression is zero or undefined, and testing intervals to find where the expression is positive.
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02:58Rationalizing Denominators
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