Question

The complete set of values of $x$ that satisfies the inequality $x^{{\log }_{5}x}>5$ is

• A. $x\in (0,\infty )$
• B. $x\in (0,\frac{1}{5})\cup (5,\infty )$
• C. $x\in (1,\infty )$
• D. $x\in (1,2)$

Answer 1

The correct option is B $x\in (0,\frac{1}{5})\cup (5,\infty )$

$x^{{\log }_{5}x}>5$, where $x>0$ ...(1)
Take log both sides,
$\Rightarrow {\log }_{5}x\left(\log x\right)>\log 5$
$\Rightarrow {\left(\log x\right)}^2>{\left(\log 5\right)}^2$
$\Rightarrow \log x>\log 5or\log x<-\log 5$
$\Rightarrow x\in (-\infty ,\frac{1}{5})\cup (5,\infty )$ ...(2)
From (1) & (2)
$\Rightarrow x\in (0,\frac{1}{5})\cup (5,\infty )$
Ans: B