Question

$f\left(x\right)={\log }_{\sqrt{2}}(2-{\log }_2(16si{n}^{2}x+1\left)\right)$. What is its range?

Answer 1

$f\left(x\right)={\log }_{\sqrt{2}}(2-{\log }_2(16{\sin }^{2}x+1))$

$={\log }_{\sqrt{2}}(2-\log \left(16\right[0,1]+1))$ (${\sin }^{2}x$ is continuous increasing function)

$={\log }_{\sqrt{2}}(2-[0,{\log }_{2}17])$ (${\log }_2$ is continuous increasing function)

${\log }_{\sqrt{2}}(2-{\log }_2\left(4\right),2)$ (${\log }_{\sqrt{2}}$ is also increasing function, $\sqrt{2}>1$)

$=(\infty ,\frac{1}{2}]$ (as $2-{\log }_{2}17<0$ taking ${\log }_2\left(4\right)=0$)