Question

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

Answer 1

$x<0,So\left|x\right|=-x$

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

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

$z=\sqrt{z}$

$z=1,0$

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

$x=-1$

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

$x=-10$

$thereforetwosolutionsarepossible.$