Question

Solve: $x^{\left(\frac{1}{{\log }_{10}x}\right)}.{\log }_{10}x<1$

• A. $0<x<10$
• B. $0<x<{10}^{10}$
• C. $0<x<1$
• D. $0<x<{10}^{\frac{1}{10}}$

Answer 1

The correct option is D $0<x<{10}^{\frac{1}{10}}$

$\Rightarrow \left(x\frac{{\log }_{10}10}{{\log }_{10}x}\right)-{\log }_{10}x<1(1={\log }_{10}10)$

$\Rightarrow x^{{\log }_{x}10}-{\log }_{10}x<1(\frac{{\log }_{a}b}{{\log }_{a}c}={\log }_{c}b)$

$\Rightarrow 10-{\log }_{10}x<1(\because a^{{\log }_{a}b}=b)$

$\Rightarrow {\log }_{10}{x}^{10}<1(\because a{\log }_{c}b={\log }_c{b}^a)$

$\Rightarrow x^{10}<{10}^1$ (Taking antilog)

$\Rightarrow x<{10}^{\frac{1}{10}}$

and in ${\log }_{10}x,x>0$

$\Rightarrow 0<x<{10}^{\frac{1}{10}}$