Answer
$\{-6,6\}$
Work Step by Step
We have to find the value(s) of $x$ so that:
$$\log_{5}\sqrt{x^2-9}=1.$$
Rewrite the equation in exponential form:
$$\log_{a}b=c\Rightarrow b=a^c$$
$$\sqrt{x^2-9}=5$$
Square both sides:
$$x^2-9=25$$
$$x^2=36$$
$$x_{1}=6 \text{ and } x_{2}=-6$$
Check if the solutions are valid:
$$\log_5\sqrt{(-6)^2-9}=\log_5\sqrt{25}=\log_5 5=1\checkmark$$
$$\log_5\sqrt{6^2-9}=\log_5\sqrt{25}=\log_5 5=1\checkmark$$
The solution set is $\{-6,6\}$
