Question

Number of real solutions of the equation $\sqrt{{\log }_{10}(-x)}={\log }_{10}\sqrt{x^2}$ is :

• A. zero
• B. exactly 1
• C. exactly 2
• D. 5

Answer 1

The correct option is A zero

Given,

$\sqrt{{\log }_{10}(-x)}={\log }_{10}\left(\sqrt{x^2}\right)$

$\sqrt{{\log }_{10}(-x)}={\log }_{10}x$

${\log }_{10}(-x)={\left({\log }_{10}\left(x\right)\right)}^2$

${\log }_{10}(-1)+{\log }_{10}\left(x\right)={\left({\log }_{10}\left(x\right)\right)}^2$

let ${\log }_{10}\left(x\right)=u$

${\log }_{10}(-1)+u={\left(u\right)}^2........{\log }_{10}(-1)isnotdefined$

$u=Undefined$

${\log }_{10}\left(x\right)=Undefined:x={10}^{Undefined}$

$x={10}^{Undefined}False$

No solution for $x\in R$