Question

$\sqrt{{\log }_{1/2}\left({\log }_2\right(x^2+4x+5\left)\right)}$.

Answer 1

$\sqrt{{\log }_{1/2}\left({\log }_2\right(x^2+4x+5\left)\right)}$

$\because$ exp $=x^2+4x+5$

$=(x+2{)}^2+1$

$\therefore$ Range $=(1,\infty )$

$\therefore$ Range of ${\log }_2(x^2+4x+5)$ is $[0,\infty )$

And

$\therefore$ Range of ${\log }_{\frac{1}{2}}\left({\log }_2\right(x^2+4x+5\left)\right)$ is always negative.

$\therefore$ No solution of the problem exists.