Question

Solution set of the inequality:
$\sqrt{{13}^x-5}\le \sqrt{2({13}^x+12)}-\sqrt{{13}^x+5}$ is

• A. $(-\infty ,-5)$
• B. $(5,\infty )$
• C. $[{\log }_{13}5,1]$
• D. $[0,{\log }_{13}5]$

Answer 1

The correct option is C $[{\log }_{13}5,1]$

$\sqrt{{13}^x-5}\le \sqrt{2({13}^x+12)}-\sqrt{{13}^x+5}$
Above equation is valid when ${13}^x-5\ge 0\Rightarrow {13}^x\ge 5$
Taking log on base 13
$\Rightarrow x\ge {\log }_{13}5$ .....(1)

On substituting $t={13}^x$, we get
$\sqrt{t-5}+\sqrt{t+5}\le \sqrt{2(t+12)}$
$\Rightarrow t-5+t+5+2\sqrt{t^2-25}\le 2t+24$
$\Rightarrow \sqrt{t^2-25}\le 12$
$\Rightarrow t^2\le 169$
$\Rightarrow t\le 13$
$\Rightarrow {13}^x\le 13$
$\Rightarrow x\le 1$ ....(1)
From (1) & (2), we get
$x\in [{\log }_{13}5,1]$

Ans: C