12. Find the domain of f (x) = ln x- 5 x+7

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) = \
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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