Question

The range of values of $x$ which satisfy the inequation $x^{\frac{1}{{\log }_{10}x}}\cdot {\log }_{10}x<1$ is
$(x\ne 1)$

• A. $x\in (0,10)-\left\{1\right\}$
• B. $x\in (0,{10}^{10})-\left\{1\right\}$
• C. $x\in (0,1)$
• D. $x\in (0,{10}^{\frac{1}{10}})-\left\{1\right\}$

Answer 1

The correct option is D $x\in (0,{10}^{\frac{1}{10}})-\left\{1\right\}$

From the given inequation, it is clear that $x>0$
Given: $x\ne 1$, then inequation can be written as
$x^{{\log }_{x}10}\cdot {\log }_{10}x<1$
$\Rightarrow {\log }_{10}x<\frac{1}{10}$
$\Rightarrow 0<x<{10}^{\frac{1}{10}}$