Logarithmic function for complex numbers

Question

For the number

$z=i^{i^i}$

can I take a log on both sides and write it as $\log(z)=i^{i}\log(i)$?

I know that we can write $\log(e^{iθ})=iθ\log(e)$, but I'm no sure if we can do that when the base is non real. Is this step valid for a non real base?

Answer 1

I don't think that will work. But you could simplify is by using the definition from top to down. First, compute $w=i^i$ and then $z=i^w$.

By definition $$i^i=e^{i\log(i)}$$ where $\log(i)=\ln(|i|)+Arg(i)i=\frac12\pi i$. So we get $w=i^i=e^{-\frac12\pi}$. Next, $$z=i^w=e^{w\log(i)}=e^{\frac12w\pi i}=e^{i\frac12\pi e^{-\frac12\pi}}.$$