Question

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

• A. zero
• B. exactly $1$
• C. exactly $2$
• D. $4$

Answer 1

Answer: (C)

exactly $2$

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

$\Rightarrow \sqrt{{\log }_{10}(-x)}={\log }_{10}|x|$ for ${\log }_{10}(-x)xϵ(-\infty ,0)$

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

[Here $x$ should be less then $0$]

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

$\Rightarrow {\log }_{10}{(-x)}^2-{\log }_{10}(-x)=0$

$\Rightarrow {\log }_{10}(-x)({\log }_{10}(-x)-1)=0$

$\Rightarrow {\log }_{10}(-x)=0$

$\Rightarrow (-x)=1$

$\Rightarrow x=-1$

And, $\Rightarrow {\log }_{10}(-x)=1$

$\Rightarrow (-x)=10$

$\Rightarrow x=-10$

The equation have exactly $2$ solution.