Construct a branch of $\sqrt{\log{z}}$ on the upper half-plane ${z\in C:Im(z)\geq0}$. Justify your answer

Question

I can't quite figure out where I've gone wrong, my understanding is it should be something like: $$\sqrt{\ln\vert z\vert+iArg(z)}$$ where $Arg(z)$ is the principle argument for z.