$2 y=\log _{a} x(x>1, a>1)$ graph has three points $A, B, C$, with horizontal coordinates $m, m+2$, $m+4$ respectively. If the area of $\triangle A B C$ is $S=f(m)$, then the range of $f(m)$ is
$$
\begin{array}{l}
2\left(0, \log _{a} 3-\frac{1}{2} \log _{a} 5\right) . \\
f_{(m)}=\frac{1}{2}\left(\log _{a} m+2 \log _{a}(m+2)+\log _{a}(m+4)\right)-\left(\log _{a} m+\log _{a}(m+4)\right) \\
\quad=\frac{1}{2} \log _{a} \frac{(m+2)^{2}}{m(m+4)}=\frac{1}{2} \log _{a}\left(1+\frac{4}{m(m+4)}\right)
\end{array}
$$

is a decreasing function of $m$, as $m \rightarrow 1$, $f(m) \rightarrow \log _{a} 3-\frac{1}{2} \log _{a} 5$. As $m \rightarrow+\infty$, $f(m) \rightarrow 0$.