College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter5: Exponential And Logarithmic Functions
Section5.3: Logarithmic Functions And Their Graphs
Problem 140E
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![### Problem 12: Domain of a Logarithmic Function
**Problem Statement:**
Find the domain of \( f(x) = \ln \left( \frac{x - 5}{x + 7} \right) \).
---
**Explanation:**
To determine the domain of the function \( f(x) = \ln \left( \frac{x - 5}{x + 7} \right) \), we need to understand the conditions under which the logarithm is defined.
The natural logarithm function, \( \ln(y) \), is defined only for \( y > 0 \). Therefore, the argument inside the logarithm \( \left( \frac{x - 5}{x + 7} \right) \) must be greater than zero.
So, we need to solve the inequality:
\[ \frac{x - 5}{x + 7} > 0 \]
There are two scenarios where a fraction is positive:
1. Both the numerator and the denominator are positive.
2. Both the numerator and the denominator are negative.
Let's consider these cases:
1. **Case 1: Both numerator and denominator are positive:**
\[
\begin{cases}
x - 5 > 0 \\
x + 7 > 0
\end{cases}
\]
Solving these inequalities:
\[
\begin{cases}
x > 5 \\
x > -7
\end{cases}
\]
The common solution here is \( x > 5 \).
2. **Case 2: Both numerator and denominator are negative:**
\[
\begin{cases}
x - 5 < 0 \\
x + 7 < 0
\end{cases}
\]
Solving these inequalities:
\[
\begin{cases}
x < 5 \\
x < -7
\end{cases}
\]
The common solution here is \( x < -7 \).
Summarizing these results, the values of \( x \) that satisfy the fraction \(\frac{x - 5}{x + 7} > 0 \) are:
\[ (-\infty, -7) \cup (5, \infty) \]
Hence, the domain of the function \( f(x) = \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F432d0c31-f5e9-405a-bdb3-3bc4909f09a2%2F4b5a3628-595a-459a-b624-7e576c58c8a3%2Fom74fce_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 12: Domain of a Logarithmic Function
**Problem Statement:**
Find the domain of \( f(x) = \ln \left( \frac{x - 5}{x + 7} \right) \).
---
**Explanation:**
To determine the domain of the function \( f(x) = \ln \left( \frac{x - 5}{x + 7} \right) \), we need to understand the conditions under which the logarithm is defined.
The natural logarithm function, \( \ln(y) \), is defined only for \( y > 0 \). Therefore, the argument inside the logarithm \( \left( \frac{x - 5}{x + 7} \right) \) must be greater than zero.
So, we need to solve the inequality:
\[ \frac{x - 5}{x + 7} > 0 \]
There are two scenarios where a fraction is positive:
1. Both the numerator and the denominator are positive.
2. Both the numerator and the denominator are negative.
Let's consider these cases:
1. **Case 1: Both numerator and denominator are positive:**
\[
\begin{cases}
x - 5 > 0 \\
x + 7 > 0
\end{cases}
\]
Solving these inequalities:
\[
\begin{cases}
x > 5 \\
x > -7
\end{cases}
\]
The common solution here is \( x > 5 \).
2. **Case 2: Both numerator and denominator are negative:**
\[
\begin{cases}
x - 5 < 0 \\
x + 7 < 0
\end{cases}
\]
Solving these inequalities:
\[
\begin{cases}
x < 5 \\
x < -7
\end{cases}
\]
The common solution here is \( x < -7 \).
Summarizing these results, the values of \( x \) that satisfy the fraction \(\frac{x - 5}{x + 7} > 0 \) are:
\[ (-\infty, -7) \cup (5, \infty) \]
Hence, the domain of the function \( f(x) = \
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