Find the domain on which $\log(z^2+9)$ is analytic. I have the following function
$$f(z)=\log(z^2+9)$$
I need to find the set on which this function is analytic.
So far I know that $\log(z)$ is analytic in $D= \left \{ z \in \mathbb{C} | z \notin (-\infty, 0] \right \}$.
So, I understand that for my function $D= \left \{ z \in \mathbb{C} | z^2+9 \notin (-\infty, 0] \right \} = \left \{ z \in \mathbb{C} | z^2 \notin (-\infty, -9] \right \}$.
How should I continue?

$\left \{ z \in \mathbb{C} | z^2 \notin (-\infty, -9] \right \}$ is a perfectly good specification of the set, so you could reasonably stop here. However, if you'd prefer an interval form, note that $z^2 \in (-\infty, -9]$ means $z$ is purely imaginary and has imaginary part at least $3$ or no more than $-3$. Thus $D$ is the union of two rays:
$$
D = \{z\in \mathbb C | iz\notin (-\infty, -3]\} \cup \{z\in \mathbb C | iz\notin[3,\infty)\} 
$$
