Problem 110 - College Algebra Essentials

Question

Find the domain of each logarithmic function. $$f(x)=\ln \left(x^{2}-4 x-12\right)$$

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\ln \left(x^{2}-4 x-12\right)\) is \(x \in (-\infty, -2) \cup (6, \infty)\).

Step 1: Explore the Logarithm Rule

The logarithmic function \(f(x)=\ln \left(x^{2}-4 x-12\right)\) must be greater than zero (0) because the log of a negative number or 0 is undefined. Therefore, to ensure the function is defined, we need to find when \(x^{2}-4 x-12 > 0\).

Step 2: Factorize the Polynomial

We factorize the equation \(x^{2}-4 x-12 > 0\) to find the critical points. Upon factorizing, we get \((x - 6)(x + 2) > 0\)

Step 3: Solve for x

To find the critical points, we set each factor greater than zero and solve for \( x \):1) \( x - 6 > 0 \) which leads to \(x > 6\).2) \( x + 2 > 0 \) which leads to \(x > -2\).

Step 4: Test Intervals

Now we test the intervals to find the solution. Take a number from each of the intervals \((-\infty, -2), (-2, 6), (6, \infty)\). For \((-\infty, -2)\), choose \(x = -3\). For \((-2, 6)\), choose \(x = 0\). For \((6, \infty)\), choose \(x = 7\). All of these values should be substituted in \((x - 6)(x + 2) > 0\) to find out which makes the inequality true.

Step 5: Find the Domain

From the test of intervals we find that the intervals which satisfy the inequality are \((-\infty, -2)\) and \((6, \infty)\). Hence, the domain of the function \(\ln \left(x^{2}-4 x-12\right)\) will be \(x \in (-\infty, -2) \cup (6, \infty)\).