Question

Solve the following equation for x
$x^{\frac{2}{3}\left[\right(lo{g}_{2}x{)}^2+lo{g}_{2}x-5/4]}=\sqrt{2}$

Answer 1

$x^{\left(2/3\right)\left[\right(lo{g}_{2}x{)}^2+lo{g}_{2}x-5/4]}=\sqrt{2}$
Put $lo{g}_{2}x=t$ $\therefore x=2^t$ ........(1)
$\frac{2}{3}[t^2+t-\frac{5}{4}]=lo{g}_x\sqrt{2}=\frac{1}{2}lo{g}_{x}2=\frac{1}{2t}.$
$4t(t^2+t-\frac{5}{4})=3$
or $4t^2+4t^2-5t-3=0$
or $(t-1)(4t^2+8t+3)=0$
or $(t-1)(2t+3)(2t+1)=0$
$\therefore t=-3/4,-1/2,1=lo{g}_{2}x$
$\therefore x=2^{-3/4},2^{-1/2},2^1=\frac{1}{2\sqrt{(}2)},\frac{1}{\sqrt{(}2)},2$