Problem 112 - College Algebra Essentials

Question

Find the domain of each logarithmic function. $$f(x)=\log \left(\frac{x-2}{x+5}\right)$$

Step-by-Step Solution

Answer

The domain of the given function is \( (2, ∞)\cup(-5, 2) \).

Step 1: Identify the Expression within the Logarithm

The expression within the logarithm is \(\frac{x-2}{x+5}\). This expression must be greater than zero.

Step 2: Solve the Inequalities

Set up two separate inequalities to solve: 1. \(x - 2 > 0\), which simplifies to \(x > 2\)2. \(x + 5 \neq 0\), which simplifies to \(x \neq -5\)

Step 3: Combine the Results

Combine the two results from above. The solution must satisfy both conditions. So, the domain of \(f\) is all real numbers greater than 2, excluding -5. This can be written in interval notation as \((2, ∞)\cup(-5, 2)\).

Step 4: Verify the Solution

To verify, substitute a number from each interval into the original function. You'll see that it will produce a valid output, meaning the original function is defined at those points.