Example 1 Solve the equation: $\lg (3-x)-\lg (3+x)=\lg (1-x)-\lg (2 x+1)$.
Solve $(3-x)(2x+1)=(3+x)(1-x)$, we get $x=7$ or $x=0$. Upon verification, $x=7$ is an extraneous root of the original equation, and $x=0$ is a root of the original equation.