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Chapter 6: Problem 39

Find the domain of each function. $$ f(x)=\ln (x-3) $$

Short Answer

Expert verified
The domain is \[ (3, \infty) \].

Step by step solution

01

Understand the properties of the natural logarithm function

The natural logarithm function, \( \ln(x) \), is only defined for positive values of \ x \. Hence, any input to \( \ln(x) \)must be greater than zero.
02

Set the argument of the logarithm greater than zero

Since in this case the argument is \( x - 3 \, \ x - 3 \) must be greater than 0, so we need to solve the inequality \( x - 3 > 0 \).
03

Solve the inequality

Add 3 to both sides of the inequality: \[ x - 3 > 0 \] becomes \[ x > 3 \].
04

Write the domain of the function

The solution to the inequality \[ x > 3 \] gives us the domain. Therefore, the domain of \( f(x) = \ln(x - 3) \) is \[ (3, \infty) \].

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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 \( \ln(x) \), is a fundamental mathematical function with base \( e \). Here, \( e \) is Euler's number, approximately equal to 2.71828. The natural logarithm has several key properties:
  • It is only defined for positive values of \( x \).
  • \( \ln(1) = 0 \), because \( e^0 = 1 \).
  • \( \ln(e) = 1 \), because \( e^1 = e \).
  • It increases slowly as \( x \) increases.
In the context of functions, if we have \( \ln(g(x)) \), then \( g(x) \) must be positive. This property is critical for determining the domain of logarithmic functions as it restricts the acceptable input values.
Inequality
Inequalities are statements that compare two values or expressions. They use symbols like <, >, \( \leq \), and \( \geq \). In this exercise, we are dealing with the inequality \( x - 3 > 0 \). Understanding how to solve such inequalities is essential:
1. \( x - 3 > 0 \) means \( x \) must be greater than 3.
2. To solve \( x - 3 > 0 \), we isolate \( x \):
  • Add 3 to both sides: \( x > 3 \).
Solving inequalities helps us find the set of values for which an expression holds true. This is particularly useful when determining the domain of functions, as it tells us which input values produce defined (i.e., non-negative) outputs.
Function Domain
The domain of a function refers to all possible input values (\( x \)-values) for which the function is defined. For logarithmic functions like \( \ln(x - 3) \), the argument (\( x - 3 \)) must be greater than 0 for the function to be defined:
  • The function \( f(x) = \ln(x - 3) \) is only defined when \( x - 3 > 0 \).
  • From solving the inequality \( x - 3 > 0 \), we get \( x > 3 \).
Thus, the domain of \( f(x) \) is all \( x \)-values greater than 3. In interval notation, this is expressed as \( (3, \infty) \). Understanding the domain helps to identify where the function will produce valid outputs. Therefore, the domain of the given function \( f(x) = \ln(x - 3) \) is \( (3, \infty) \).

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