Example 6 Given that for all real numbers $x$, the inequality $\left[\left(\log _{3} m\right)^{2}-\log _{3}\left(27 m^{2}\right)\right] x^{2}-\left(\log _{3} m-3\right) x-1<0$ always holds, find the range of real numbers $m$.
(1) If $\left(\log _{3} m\right)^{2}-\log _{3}\left(27 m^{2}\right)=0, \log _{3} m=3$ or -1
If $\log _{3} m=-1$, the original inequality is $4 x-1<0$, which does not meet the requirement;
If $\log _{3} m=3$, the original inequality is $-1<0$, which is obviously true for any $x \in \mathbf{R}$, solving gives $m=27$
(2) If $\left(\log _{3} m\right)^{2}-\log _{3} 27 m^{2} \neq 0$, to make the original inequality true for any $x \in \mathbf{R}$, then it should be:
$$
\left\{\begin{array}{l}
\left(\log _{3} m\right)^{2}-\log _{3}\left(27 m^{2}\right)<0 \\
\Delta=\left(\log _{3} m-3\right)^{2}+4\left[\left(\log _{3} m\right)^{2}-\log _{3}\left(27 m^{2}\right)\right]<0
\end{array}\right.
$$

Solving gives $-\frac{1}{5}<\log _{3} m<3,3^{-\frac{1}{5}}<m<27$
In summary, the range of real number $m$ is $\left(3^{-\frac{1}{5}}, 27\right]$