Problem 59 - College Algebra

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 all real numbers except 2.

Step 1: Identify the Argument

The argument of the logarithmic function, \(f(x) = \ln (x-2)^{2}\) is \((x-2)^{2}\). This is what we will analyze to find the domain.

Step 2: Set Up the Inequality

For the logarithmic function to exist, whatever is inside the logarithm (the argument) must be greater than zero. We set up the inequality: \((x-2)^{2} > 0\). It's important to note that since it's squared, the value inside the bracket can actually be either negative or positive, it doesn't matter because the square of a real number is always greater than or equal to zero, except for zero itself.

Step 3: Solve the Inequality

As the square of any real number, except zero, is always greater than zero, the solution to the inequality will be the set of all real numbers except 2. Therefore, the domain is all real numbers except 2. We remove 2 from the domain, because \((2-2)^2=0\) and you can't take the log of zero.