$\Rightarrow f (0)= e$
$f ^{\prime}( x )+ f ( x ) \sqrt{1-(\ln f ( x ))^2}=0$
$f ( x )= y$
$\frac{ dy }{ dx }=- y \sqrt{1-(\ln y )^2}$
$\int \frac{ dy }{ y \sqrt{1-(\ln y )^2}}=-\int dx$
$\text { Put } \ln y = t$
$\int \frac{ dt }{\sqrt{1-t^2}}=- x + C$
$\sin ^{-1} t=-x+C \Rightarrow \sin ^{-1}(\ln y )=- x + C$
$\sin ^{-1}(\ln f ( x ))=- x + C$
$f(0)= e$
$\Rightarrow \frac{\pi}{2}= C$
$\Rightarrow \sin ^{-1}(\ln f ( x ))=- x +\frac{\pi}{2}$
$\Rightarrow \sin ^{-1}\left(\ln f \left(\frac{\pi}{6}\right)\right)=\frac{-\pi}{6}+\frac{\pi}{2}$
$\Rightarrow \sin ^{-1}\left(\ln f \left(\frac{\pi}{6}\right)\right)=\frac{\pi}{3}$ $\Rightarrow \ln f \left(\frac{\pi}{6}\right)=\frac{\sqrt{3}}{2}, \text { we need }\left(6 \times \frac{\sqrt{3}}{2}\right)^2=27$
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