# Chapter 6 - Section 6.4 - Logarithmic Functions - 6.4 Assess Your Understanding: 43

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 $(10, +\infty)$.

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