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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblema 75
Capitolo 5, Problema 75

Find the domain of each logarithmic function. f(x) = log5(x+4)

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Recall that the domain of a logarithmic function \( f(x) = \log_b(g(x)) \) consists of all values of \( x \) for which the argument \( g(x) \) is positive, because the logarithm of zero or a negative number is undefined.
Identify the argument of the logarithm in the given function: \( f(x) = \log_5(x+4) \). Here, the argument is \( x + 4 \).
Set up the inequality to find where the argument is positive: \( x + 4 > 0 \).
Solve the inequality for \( x \): subtract 4 from both sides to get \( x > -4 \).
Conclude that the domain of \( f(x) = \log_5(x+4) \) is all real numbers \( x \) such that \( x > -4 \).

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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 domain is restricted because the argument inside the log must be positive. Identifying the domain involves finding all x-values that make the expression inside the logarithm greater than zero.
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Properties of Logarithmic Functions

A logarithmic function log_b(x) is defined only for positive arguments x > 0, where b is the base and b > 0, b ≠ 1. This means the expression inside the log must be strictly greater than zero to avoid undefined values or complex numbers.
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Inequalities and Solving for Domain

To find the domain of f(x) = log_5(x+4), solve the inequality x + 4 > 0. This involves basic algebraic manipulation to isolate x, resulting in the domain expressed as an interval. Understanding how to solve inequalities is essential for determining valid input values.
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