Questions: Find the domain of the logarithmic function. f(x)=ln(x^2-25) The domain of f(x) is . (Type your answer in interval notation.)

Find the domain of the logarithmic function. f(x)=ln(x^2-25) The domain of f(x) is . (Type your answer in interval notation.)
Transcript text: Find the domain of the logarithmic function. \[ f(x)=\ln \left(x^{2}-25\right) \] The domain of $f(x)$ is $\square$ $\square$. (Type your answer in interval notation.)
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Solution

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Solution Steps

To find the domain of the logarithmic function \( f(x) = \ln(x^2 - 25) \), we need to determine the values of \( x \) for which the expression inside the logarithm is positive. This means solving the inequality \( x^2 - 25 > 0 \).

Solution Approach
  1. Set up the inequality \( x^2 - 25 > 0 \).
  2. Solve for \( x \) by factoring the quadratic expression.
  3. Determine the intervals where the inequality holds true.
  4. Express the solution in interval notation.
Step 1: Set Up the Inequality

To find the domain of the function \( f(x) = \ln(x^2 - 25) \), we need to solve the inequality: \[ x^2 - 25 > 0 \]

Step 2: Factor the Expression

The expression can be factored as: \[ (x - 5)(x + 5) > 0 \]

Step 3: Determine the Intervals

To solve the inequality \( (x - 5)(x + 5) > 0 \), we find the critical points where the expression equals zero: \[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \] \[ x + 5 = 0 \quad \Rightarrow \quad x = -5 \] These critical points divide the number line into three intervals: \( (-\infty, -5) \), \( (-5, 5) \), and \( (5, \infty) \).

Step 4: Test the Intervals

We test each interval to determine where the product is positive:

  • For \( x < -5 \) (e.g., \( x = -6 \)): \( (-6 - 5)(-6 + 5) = (-11)(-1) > 0 \)
  • For \( -5 < x < 5 \) (e.g., \( x = 0 \)): \( (0 - 5)(0 + 5) = (-5)(5) < 0 \)
  • For \( x > 5 \) (e.g., \( x = 6 \)): \( (6 - 5)(6 + 5) = (1)(11) > 0 \)

Thus, the solution to the inequality is: \[ (-\infty, -5) \cup (5, \infty) \]

Final Answer

The domain of \( f(x) \) is: \[ \boxed{(-\infty, -5) \cup (5, \infty)} \]

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