I really don't understand this:

The set of values of $\displaystyle log(i^{1/2})$ is $\displaystyle (n+1/4)\pi i$, with $\displaystyle n \in Z$, and that the same is true of $\displaystyle \frac{1}{2} log(i)$, but...

The set of values of $\displaystyle log(i^2)$ isnotthe same as the set of values of $\displaystyle 2 log(i)$.