To find the domain of the function \( g(x) = \ln((x - 14)^2) \), we need to determine the values of \( x \) for which the argument of the logarithm is positive.
The argument of the logarithm is \( (x - 14)^2 \). Since squaring any real number (whether it is positive, negative, or zero) results in a non-negative value, \( (x - 14)^2 \) is always greater than or equal to 0.
However, the logarithm function \( \ln(y) \) is only defined for \( y > 0 \). Therefore, we require:
\[ (x - 14)^2 > 0 \]
This inequality is satisfied for all \( x \) except where \( (x - 14)^2 = 0 \), which occurs when:
\[ x - 14 = 0 \implies x = 14 \]
Thus, the function \( g(x) = \ln((x - 14)^2) \) is undefined when \( x = 14 \).
Consequently, the domain of the function excludes \( x = 14 \) and includes all other real numbers.
In interval notation, the domain can be expressed as:
\[ (-\infty, 14) \cup (14, \infty) \]