Problem 60 - College Algebra

Question

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

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\ln (x-7)^{2}\) are all real numbers except \(x = 7\).

Step 1: Identify the argument of the logarithm

The argument of the logarithm in the function \(f(x)=\ln (x-7)^{2}\) is \((x-7)^2\). This is the value we need to be greater than zero since logarithms are undefined for arguments less than or equal to zero.

Step 2: Find the domain of the argument

We must find the values of \(x\) such that \((x-7)^2>0\). However, any square of a real number is always non-negative (i.e., is either positive or zero). Therefore, the terms \((x-7)^2\) yields values greater than or equal to zero.

Step 3: Exclude zero from the domain

Because the argument of a logarithm cannot be equal to zero, we must exclude the value \(x=7\) from the domain since that causes \((x-7)^2\) to equal zero. Hence, \(x\) can be any real number other than 7.