Let f(x)=( a^ ^4 x a^ 2 ^2 x )^3 · (a^ 5 ^3 x · a^ 5 x ) and b=( 5...

$\because \quad f(x)=a^{3 \sin ^4 x+5 \sin ^3 x-6 \sin ^2 x+5 \sin x} \quad$ and $b=3 \quad(\sin x\gt0)$ $\therefore \quad$\left(f(x) \cdot a^3\right)^{\operatorname{cosec}^2 x}=a^{3 \sin ^2 x+5 \sin x-6+\frac{5}{\sin x}+\frac{3}{\sin ^2 x}} =\mathrm{a}^{3\left(\sin ^2 x+\frac{1}{\sin ^2 x}\right)+5\left(\sin x+\frac{1}{\sin x}\right)-6} \because \quad$min. value of$3\left(\sin ^2 x+\frac{1}{\sin ^2 x}\right)+5\left(\sin x+\frac{1}{\sin x}\right)-6 \quad($for$\sin x\gt0) =3(2)+5(2)-6=10$($\because \sin \mathrm{x}\gt0$and the value is minimum at$\sin \mathrm{x}=1$)$\Rightarrow \log _a(\sin x)=0