If $z=i^{2i}$ then $|z| =?$ I am given a complex number $z=i^{2i}$ so what would be $|z|$?
Firstly I took $\log$
$$
\log(z)=2i\log i\\
$$
thus
$$
\log(z)=2i\log(e^{-π/2}) = -i\pi;
$$
then inverting the log
$$
z=e^{ -i\pi}
$$
implying
$$ |z| =1.$$
Is this correct?

All of the values of $i^{2i}$ are real. The possible values are
 $$i^{2i} = e^{2i\cdot\log i}=e^{2i\cdot(\frac{\pi}{2}i+2k\pi i)}
=e^{-\pi -4k\pi}=e^{-(4k+1)\pi}
$$
which are all positive, so they are all possible values of $|z|$. 
Note that none of these is $1$.
