The number of points on the real line where the function f(x) =${\log }_{{\left|x^2-1\right|}^{ }}\left|x-3\right|$  is not defined is

${\text{log}}_{{\left|x^2-1\right|}^{ }}\left|x-3\right| is defined when x-3\ne 0 and$

$$\begin{gathered} \left|x^2-1\right| >0 and \left|x^2-1\right|\ne 1 \\ x\ne 3, x^2-1\ne 0 , x\ne 1 or -1 \end{gathered}$$
$$\begin{gathered} \left|x^2-1\right|\ne 1 \Rightarrow x\ne -\sqrt{2 }or \sqrt{2} and x\ne 0 \\ \therefore the points at which the function is not defined are x=0,1,-1,-\sqrt{2 }, \sqrt{2},3 \end{gathered}$$