Thank you for reporting, we will resolve it shortly
Q.
The equation $\sqrt{1+\log \sqrt{27}} \log _2 x +1=0$ has
Continuity and Differentiability
Solution:
$\left(\sqrt{1+\frac{3}{2 \log _3 x}}\right) \log _3 x+1=0$
$\text { let } \log _3 x = y $
$\left(\sqrt{1+\frac{3}{2 y}}\right) y=-1 \Rightarrow\left(1+\frac{3}{2 y}\right)=\frac{1}{y^2} \Rightarrow \frac{2 y+3}{2 y}=\frac{1}{y^2} $
$2 y ^2+3 y -2=0 \Rightarrow 2 y ^2+4 y - y -2=0 \Rightarrow( y +2)(2 y -1)=0 $
$y =1 / 2 \text { or } y =-2 \Rightarrow x =3^{1 / 2} \text { (rejected) or } x =1 / 9$