(c) ${\log _{10}}({x^2} – 2x – 2) \le 0$…..$(i)$

For logarithm to be defined,

${x^2} – 2x – 2 > 0$

$ \Rightarrow $ ${(x – 1)^2} > 3$

==> $x – 1 < – \sqrt 3 $ or$x – 1 > \sqrt 3 $

==> $x < 1 – \sqrt 3 $ or $x > 1 + \sqrt 3 $

i.e., $x < – (\sqrt 3 – 1)$ or $x > (\sqrt 3 + 1)$

Now from $(i),$ ${x^2} – 2x – 2 \le 1$

==> ${x^2} – 2x – 3 \le 0$

==> $(x – 3)\,(x + 1) \le 0$ $ \Rightarrow $ $ – 1 \le x \le 3$

$\therefore $ $x \in [ – 1,\, – (\sqrt 3 – 1)\,[\, \cup \,]\,\,\sqrt 3 + 1,\,3]$.

i.e., $x \in [ – 1,\,1 – \sqrt 3 )\,\, \cup (1 + \sqrt 3 ,\,3)$.