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Use a graphing utility and the change-of-base property to graph each function. y = log3 x
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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 79
Find the domain of each logarithmic function. f(x) = ln (x-2)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 strictly greater than zero, so we need to find where \( (x-2)^2 > 0 \).
Set up the inequality \( (x-2)^2 > 0 \) and analyze it. Since a square of any real number is always non-negative, \( (x-2)^2 \geq 0 \) for all \( x \), but it equals zero when \( x = 2 \).
Because the logarithm is undefined at zero, exclude \( x = 2 \) from the domain. Therefore, the domain includes all real numbers except \( x = 2 \).
Express the domain in interval notation as \( (-\infty, 2) \cup (2, \infty) \).
Summarize that the domain of \( f(x) = \ln((x-2)^2) \) is all real numbers except \( x = 2 \), because the argument of the logarithm must be positive.
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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 to values that make the argument inside the logarithm positive, since the logarithm of zero or a negative number is undefined.
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3:51
Domain Restrictions of Composed Functions
Properties of Logarithmic Functions
A logarithmic function, such as ln(x), is only defined for positive arguments. This means the expression inside the logarithm must be greater than zero. Understanding this property is essential to determine the domain by setting the argument greater than zero and solving the inequality.
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5:26Graphs of Logarithmic Functions
Solving Inequalities Involving Squares
When the argument of the logarithm is a squared expression, like (x-2)², it is always non-negative. To find the domain, you must analyze when the squared expression is strictly greater than zero, since the logarithm requires a positive argument, not zero. This involves solving inequalities and understanding the behavior of squared terms.
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06:24Solving Quadratic Equations by Completing the Square
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