Ch. 4 - Exponential and Logarithmic Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 4 - Exponential and Logarithmic Functions

Problem 109

Chapter 5, Problem 109

In Exercises 109–112, find the domain of each logarithmic function. f(x) = ln (x² - x − 2)

Verified step by step guidance

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:

x ² - x - 2 > 0

Factor the quadratic expression:

x ² - x - 2 = (x - 2)(x + 1)

Determine the critical points by setting each factor equal to zero:

x -2=0

gives

x =2

, and

x +1=0

gives

x =-1

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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Related Practice

Textbook Question

In Exercises 105–108, evaluate each expression without using a calculator. log₅ (log₂ 32)

Textbook Question

In Exercises 105–108, evaluate each expression without using a calculator. log₂ (log₃ 81)

Textbook Question

In Exercises 109–112, find the domain of each logarithmic function. f(x) = log[(x+1)/(x-5)]

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Textbook Question

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.

$\frac{l o g_{7} 49}{l o g_{7} 7} = l o g_{7} 49 - l o g_{7} 7$

Textbook Question

In Exercises 105–108, evaluate each expression without using a calculator. log (ln e)

Textbook Question

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.

$l o g_{b} (x^{3} + y^{3}) = 3 l o g_{b} x + 3 l o g_{b} y$