Example 1 Given the equation about $x$: $\lg (x-1)+\lg (3-x)=\lg (a-x)$ has two distinct real roots, find the range of values for $a$.
Analysis We will convert the logarithmic equation into a quadratic equation problem and pay attention to the domain of the function.
Solution The original equation $\Leftrightarrow\left\{\begin{array}{l}10 \\ f(1)>0 \\ f(3)>0\end{array}\right.$, i.e., $\left\{\begin{array}{l}13-4 a>0 \\ a-1>0 \\ a-3>0\end{array}\right.$, so $3<a<\frac{13}{4}$.