Answer
Domain:
$x\gt 10$ or $(10, +\infty)$.
Work Step by Step
RECALL:
The logarithmic function $f(x) = \log_a{x}$ is defined only when $x \gt 0$. Thus, its domain is $x \gt 0$.
This means that the function $f(x) = 3-2\log_5{(\frac{x}{2}-5)}$ is defined only when $\frac{x}{2}-5 \gt 0$.
Solve the inequality to obtain:
$\frac{x}{2}-5 \gt 0
\\\frac{x}{2}-5+5 \gt 0+5
\\\frac{x}{2} \gt 5$
Multiply 2 on both sides of the equation to obtain:
$2(\frac{x}{2}) \gt 2(5)
\\x \gt 10$
Thus, the domain of the given function is $x\gt 10$.
In interval notation, the domain is $(10, +\infty)$.
