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Chapter 5: Problem 113

Find the domain of the function. \(f(x)=\ln |x|\)

Short Answer

Expert verified
The domain is \( (-\infty, 0) \cup (0, \infty) \).

Step by step solution

01

Understand the Function

The function given is \( f(x) = \ln |x| \). This means the natural logarithm of the absolute value of \( x \).
02

Analyzing the Natural Logarithm Function

The natural logarithm function \( \ln(y) \) is only defined when \( y > 0 \).
03

Apply the Absolute Value

Since we have \( \ln |x| \), for \( |x| \) to be greater than 0, \( x \) must be any number except 0 (i.e., \( x eq 0 \)).
04

State the Domain

The domain of the function \( f(x) = \ln |x| \) is all real numbers except 0. In interval notation, this can be written as \( (-\infty, 0) \cup (0, \infty) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural Logarithm
The natural logarithm, denoted as \(\text{ln}(x)\), is the logarithm to the base \(e\), where \(e\) is an irrational and transcendental constant approximately equal to 2.71828. Understanding the natural logarithm is crucial because it helps us solve equations involving exponential growth and decay.
The key thing to remember about the natural logarithm is that it is only defined for positive numbers. This means you can't take the natural logarithm of zero or a negative number. So, when you see something like \(\text{ln}|x|\), it implies that the argument inside the logarithm, in this case, the absolute value of \(x\), must be greater than zero.
  • The function \( \text{ln}(x) \) itself is undefined when \( x \leq 0 \).
  • For any positive number \(y\), there exists a real number \(x\) such that \( e^x = y \).
Absolute Value
The absolute value of a number, denoted as \( |x| \), represents the distance of the number from zero on the number line, without considering the direction. For example, \( |-5| = 5 \) and \( |5| = 5 \).
When it comes to our function \( f(x) = \text{ln}|x| \), the absolute value ensures that the argument inside the natural logarithm is always positive, regardless of whether \( x \) is positive or negative.
  • If \( x \) is positive, \( |x| = x \).
  • If \( x \) is negative, \( |x| = -x \).
  • Absolute value cannot be zero or a negative number for it to be applied in the natural logarithm function.
Interval Notation
Interval notation is a way of writing subsets of the real number line. For defining the domain of a function, we use interval notation to specify all the possible values that \( x \) can take.
  • Parentheses \( () \) are used to denote that an endpoint is not included in the interval.
  • Brackets \( [] \) are used to include an endpoint in the interval.
  • The union symbol \( \cup \) is used to combine two intervals.
So, for the function \( f(x) = \text{ln}|x| \), we need to exclude zero because the logarithm function is not defined at zero. Therefore, the domain in interval notation is written as \( (-\text{inf}, 0) \cup (0, \text{inf}) \). This notation means that \( x \) can be any real number except zero, breaking the real number line into two separate intervals.

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Most popular questions from this chapter

Consider quadratic functions ( \(a\) )-( h ) that follow. Without graphing them, answer the questions below. a) \(f(x)=2 x^{2}\) b) \(f(x)=-x^{2}\) c) \(f(x)=\frac{1}{4} x^{2}\) d) \(f(x)=-5 x^{2}+3\) e) \(f(x)=\frac{2}{3}(x-1)^{2}-3\) f) \(f(x)=-2(x+3)^{2}+1\) g) \(f(x)=(x-3)^{2}+1\) h) \(f(x)=-4(x+1)^{2}-3\) Which graph has vertex \((-3,1) ?\)

Solve. $$x^{3}+6 x^{2}-16 x=0$$

Determine whether each of the following is true or false. Assume that \(a, x, M,\) and \(N\) are positive. $$\log _{a} M-\log _{a} N=\log _{a} \frac{M}{N}$$

Consider quadratic functions ( \(a\) )-( h ) that follow. Without graphing them, answer the questions below. a) \(f(x)=2 x^{2}\) b) \(f(x)=-x^{2}\) c) \(f(x)=\frac{1}{4} x^{2}\) d) \(f(x)=-5 x^{2}+3\) e) \(f(x)=\frac{2}{3}(x-1)^{2}-3\) f) \(f(x)=-2(x+3)^{2}+1\) g) \(f(x)=(x-3)^{2}+1\) h) \(f(x)=-4(x+1)^{2}-3\) Consider (d) and (e). Which graph is narrower?

In Exercises \(77-80\) : a) Find the vertex. b) Find the axis of symmetry. c) Determine whether there is a maximum or a minimum value and find that value.[ 3.3] $$G(x)=-2 x^{2}-4 x-7$$
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