Solve log_{2}(x)=sqrt{y}

$$\log _ { 2 } ( x ) = \sqrt { y }$$

Solve for x (complex solution)

$$x=2^{\sqrt{y}}$$

$$Im(\ln(2^{\sqrt{y}}))-\ln(2)Im(\sqrt{y})=0$$

Solve for y (complex solution)

$$y=\left(\log_{2}\left(x\right)\right)^{2}$$

$$\left(x=1\text{ or }arg(\ln(x))<\pi \right)\text{ and }x\neq 0$$

Solve for x

$$x=2^{\sqrt{y}}$$

$$y\geq 0$$

Solve for y

$$y=\left(\log_{2}\left(x\right)\right)^{2}$$

$$x\geq 1$$