Question

The interval of $x$ in which the inequality $5^{\frac{1}{4}{\log }_5^{2}x}>5^{\frac{1}{5}{\log }_{5}x}$ holds good

• A. $(0,5^{-2\sqrt{5}}]$
• B. $[5^{2\sqrt{5}},\infty )$
• C. both $a$ and $b$
• D. none of these

Answer 1

The correct option is D none of these

$5^{\frac{1}{4}{\log }_{5}2x}>5^{\frac{1}{5}{\log }_{5}x}$
$\Rightarrow \frac{1}{4}{\log }_{5}2x>\frac{1}{5}{\log }_{5}x\Rightarrow 5{\log }_{5}2x>4{\log }_{5}x\Rightarrow {\log }_{5}x(5{\log }_{5}x-4)>0$
$\Rightarrow {\log }_{5}x<0$ or ${\log }_{5}x>\frac{4}{5}$
$\Rightarrow x<5^0$ or $x>5^{\frac{4}{5}}$