Thank you for reporting, we will resolve it shortly
Q.
The number of solution(s) of the equation $\log _3\left(x^2-3 x-3\right)=\sqrt{\log _{\frac{1}{2}}\left(1+\sqrt{x^2-1}\right)}$ is (are)
Continuity and Differentiability
Solution:
$ \log _3\left(x^2-3 x-3\right)=\sqrt{\log _{\frac{1}{2}}\left(1+\sqrt{x^2-1}\right)}$
$1+\sqrt{ x ^2-1} \geq 1 \Rightarrow \log _{\frac{1}{2}}\left(1+\sqrt{ x ^2-1}\right) \leq 0$
$\sqrt{\log _{\frac{1}{2}}\left(1+\sqrt{x^2-1}\right)}$ will be defined only when $x= \pm 1$
Now, at $x=1, x^2-3 x-3$ is negative $\Rightarrow$ LHS is not defined and at $x=-1, x^2-3 x-3=1+3-3=1 \Rightarrow$ LHS is zero
$\therefore $ at $x =-1, LHS = RHS =0$
Hence, $x=-1$ is only the solution