The number of solutions of ${\log }_4(3-2x)={\log }_2(x-2)$  is

${\log }_{2^2}(3-2x)={\log }_2(x-2)$

$\frac{1}{2}{\log }_2(3-2x)={\log }_2(x-2)$

${\log }_2\sqrt{3-2x}={\log }_2(x-2)$

$\sqrt{3-2x}=x-2$

$3-2x=x^2-4x+4$

$x^2-2x+1=0$

${(x-1)}^2=0$

$x=1$

$\text{But\hspace{0.17em}\hspace{0.17em}}ifx=1,x-2=1-2=-1<0$

$\therefore$ No solution.