Problem 79 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\ln (x-2)^{2}\) is \(x \in (-\infty , 2) \cup (2, +\infty)\

Step 1: Find the Zeroes

The first step is to find when \((x-2)^{2}=0\). This equation will give us \(x=2\).

Step 2: Checking the values

We can use a number line to check for which values of \(x\), \(f(x)\) is defined. We know that the square function \((x-2)^{2}\) is always non-negative, but at \(x=2\), \((x-2)^{2}=0\). Since \(\ln(0)\) is undefined, \(x=2\) must be excluded.

Step 3: Determine the Domain

Therefore, the domain of the function \(f(x)=\ln (x-2)^{2}\) would be all real numbers except \(x=2\)