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Chapter 4: Problem 110

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

Short Answer

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

Step by step solution

01

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 \(x^{2}-4 x-12 > 0\).
02

Factorize the Polynomial

We factorize the equation \(x^{2}-4 x-12 > 0\) to find the critical points. Upon factorizing, we get \((x - 6)(x + 2) > 0\)
03

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 \(x > -2\).
04

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.
05

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 \(x \in (-\infty, -2) \cup (6, \infty)\).

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Most popular questions from this chapter

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