Problem 80 - College Algebra Essentials

Question

Find the domain of each logarithmic function. $$f(x)=\ln (x-7)^{2}$$

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\ln (x-7)^{2}\) is \(-\infty,7)\cup(7,\infty)\).

Step 1: Identifying the Argument of the Logarithm

The argument for the logarithm in this function is \((x-7)^{2}\). This is the value we are taking the logarithm of.

Step 2: Setting Up the Inequality

As the logarithm is only defined when the argument is greater than zero, the inequality to solve for is \((x-7)^{2}>0\). We notice that squaring a quantity always results in a value that is zero or positive. Therefore, the inequality holds true for all real numbers except when \((x-7)^{2}=0\), as logarithmic functions are not defined for zero arguments.

Step 3: Solving the Inequality

Solving for \((x-7)^{2}=0\), we find that \(x=7\). Thus, the domain of the function \(f(x)=\ln (x-7)^{2}\) does not include \(x=7\), as at \(x=7\), the argument of the logarithm becomes zero and the function is undefined.

Step 4: Writing the Solution as an Interval

Since \((x-7)^{2}>0\) holds for all values of x except \(x=7\), the domain of \(f(x)=\ln (x-7)^{2}\) is the set of all real numbers except 7. We write this in interval notation as \(-\infty,7)\cup(7,\infty)\). This represents all numbers from \(-\infty\) to 7 (not inclusive), union all numbers from 7 to \(\infty\) (also not inclusive), which is the whole real line excluding 7.