Question

Solve the equation: $x^{\frac{1}{{\log }_{10}x}}.{\log }_{10}x<1$.

• 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 $0<x<{10}^{\frac{1}{10}}$

From the equation, it is clear that $x>0$
Case $1:$ If $x\ne 1$, then inequation can be written as
$x^{{\log }_{x}10}.{\log }_{10}x<1$ i.e., ${\log }_{10}x<\frac{1}{10}$
So, $0<x<{10}^{1/10}$, $x\ne 1$ ....(1)
(as base is $>1$)
Case $2:$ If $x$ $=$ $1$, we have
$1^{\infty }.$ $0=0<1$,
So, $x=1$ is also a solution.
Combining (1) and (2), we have the solution
$0<x<{10}^{\frac{1}{10}}$