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Q.
Number of real solutions of the equation $\sqrt{\log _{10}(-x)}=\log _{10} \sqrt{x^2}$ is
Continuity and Differentiability
Solution:
$ x <0 ; \sqrt{\log _{10}(- x )}=\log _{10}| x | $
$\text { squaring } \log _{10}(- x )=\log _{10}{ }^2(- x ) $
$ \log _{10}(- x )=0 \text { or } \log _{10}(- x )=1 $
$ x =-1 \text { or } x =-10]$
$| x |=- x ; \text { for } x <0$