$x^{\log_{5} x} > 5$ implies that

$x \in (2, \infty)$

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$x \in (0, \frac{1}{5}) \cup (5, \infty)$

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$x \in (1, \infty)$

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$x \in (1, 5)$

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Solution

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

Solve the given inequality:
Given inequality is
$x^{\log_{5} x} > 5$
Applying $\log_{5}$ on both sides
$\Rightarrow$ $\log_{5} (x^{\log_{5} x}) > \log_{5} 5$
$\Rightarrow$ $\log_{5} x \cdot \log_{5} x > 1 [∵ \log_{c} a^{b} = b \log_{c} a a n d \log_{a} a = 1]$
$\Rightarrow$ ${(\log_{5} x)}^{2} > 1$
$\Rightarrow$ ${(\log_{5} x)}^{2} - 1 > 0$
$\Rightarrow$ $(\log_{5} x + 1) (\log_{5} x - 1) > 0 [∵ using the identity a^{2} - b^{2} = (a + b) (a - b)]$
$\Rightarrow$ $\log_{5} x < - 1 o r \log_{5} x > 1$
$\Rightarrow$ $x < 5^{- 1} o r x > 5^{1}$
$\Rightarrow$ $x < \frac{1}{5} o r x > 5$
Also the condition $x > 0$ must be satisfied.
So, $x \in (0, \frac{1}{5}) \cup (5, \infty)$
Hence, the correct option is B.

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