Question

Find the domain of the following logarithmic function.  
g left parenthesis x right parenthesis equals ln left parenthesis left parenthesis x minus 14 right parenthesis squared right parenthesis
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Part 1
The domain is
  
enter your response here.
​(Type your answer in interval​ notation.)

Answer 1

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:

\[
(-\infty, 14) \cup (14, \infty)
\]