Question

The range of ${log}_{\sqrt{3}}[\sqrt{2}(sinx-cosx)+3]$ is

Answer 1

${\log }_{\sqrt{3}}\left[\sqrt{2}\left(\sin x-\cos x\right)+3\right]$

$-\sqrt{a^2+b^2}\le a\sin x+b\cos x\le \sqrt{a^2+b^2}$

$-\sqrt{2}\le \left(\sin x-\cos x\right)\le \sqrt{2}-2\le \sqrt{2}\left(\sin x-\cos x\right)\le 21\le \left\{\sqrt{3}\left(\sin x-\cos x\right)+3\right\}\le 5$

${\log }_{\sqrt{3}}1\le {\log }_{\sqrt{3}}\left\{\sqrt{2}\left(\sin x-\cos x\right)+3\right\}\le {\log }_{\sqrt{3}}5$

$0\le f\left(x\right)\le 2{\log }_{3}5$

$Range\in \left[0,2{\log }_{3}5\right]$