Example 2 Solve the inequality for $x$: $\log _{a}\left(4+3 x-x^{2}\right)-\log _{a}(2 x-1)>\log _{a} 2(a>0$, $a \neq 1)$.
Solving the original inequality can be transformed into $\log _{a}\left(4+3 x-x^{2}\right)>\log _{a} 2(2 x-1)$.
When $a>1$, $\left\{\begin{array}{l}2 x-1>0, \\ 4+3 x-x^{2}>0, \\ 4+3 x-x^{2}>2(2 x-1)\end{array} \Leftrightarrow\left\{\begin{array}{l}x>\frac{1}{2}, \\ -10, \\ 4+3 x-x^{2}>0, \\ 4+3 x-x^{2}\frac{1}{2}, \\ -12\end{array} \Leftrightarrow 21$ when, the solution set of the inequality is $\left(\frac{1}{2}, 2\right)$;
When $0<a<1$, the solution set of the inequality is $(2,4)$.
Key points for solving logarithmic inequalities: The key points are two: one is the domain, and the other is monotonicity.