9. Given the function defined on $\mathbf{R}^{+}$ as $f(x)=\left\{\begin{array}{ll}\left|\log _{3} x-1\right|, & 09 .\end{array}\right.$ Let $a, b, c$ be three distinct real numbers, satisfying $f(a)=f(b)=f(c)$, find the range of $a b c$.

Suppose $a<b<c$, then by the graph characteristics of $f(x)$ we have
$$
f(a)=f(b)=f(c) \Rightarrow 0<y<1 \Rightarrow 1<a<3,3<b<9,9<c<16 \text {. }
$$

And $1-\log _{3} a=\log _{3} b-1 \Rightarrow \log _{3} a+\log _{3} b=2 \Rightarrow a b=9$, so $a b c=9 c \in(81,144)$.
Conversely, take $r \in(81,144) \Rightarrow c=\frac{r}{9} \in(9,16) \Rightarrow y=4-\sqrt{c} \in(0,1)$, then there exist $1-\log _{3} a=\log _{3} b-1=y \Rightarrow a=3^{1-y} \in(1,3), b=3^{1+y} \in(3,9)$ satisfying the conditions. In summary, the range of $a b c$ is $(81,144)$.