Question

Solve the following inequality:
${\log }_a(1-x^2)⩾1.$

Answer 1

${\log }_a(1-x^2)\ge 1$

$\Rightarrow \frac{\log (1-x^2)}{\log a}\ge 1$

$\Rightarrow \log (1-x^2)\ge \log a$

$\Rightarrow 1-x^2\ge a$

$\Rightarrow x^2+(a-1)\le 0$

$\Rightarrow x^2-(1-a)\le 0$

$\Rightarrow (x+\sqrt{1-a})(x-\sqrt{1-a})\le 0$

$xϵ[-\sqrt{1-a},\sqrt{1-a}]$ (where $0<a\le 1$ because if $a<0$ then $\log a$ will not be defined).