Where is the function $\log(z)^{0.5}$ defined?

Question

I am having trouble understanding where the function $\log(z)^{0.5}$ when $f(z)=z^{0.5}$ is defined as the main branch of the square root function.

I would really appreciate your help.

Answer 1

We have that $f(z) = \exp\left(\frac12\text{Log}(z)\right)$, where $\text{Log}$ is the main branch of the logarithm. Hence,

$${\text{Log}(z)}^{0.5} = f\left(\text{Log}(z)\right) = \exp\left(\frac12\text{Log}\big(\text{Log}(z)\big)\right).$$

Since $\exp$ is defined everywhere, you must check where $\text{Log}^2 = \text{Log} \circ \text{Log}$ is well defined. Do you think you can take it from here?