Answer
$(-\infty, 7) \cup (7, +\infty)$
Work Step by Step
RECALL:
The domain of the logarithmic function $f(x) = \log_a{x}$ is $x \gt 0$.
Thus, the domain of the given function is the set of all real numbers such that:
$(x-7)^2 \gt 0$.
Note that the value of the square any number will always be positive except when the number is zero.
Thus,
The value of $(x-7)^2$ will only be 0 when $(x-7)$ itself is zero.
The value of $x-7$ is equal to zero only when $x=7$
This means that $(x-7)^2 \gt 0$ for all real numbers except 7.
Therefore, the domain of the given function is the set of all real numbers except 7.
In interval notation, this is $(-\infty, 7) \cup (7, +\infty)$.
