On normal operator logarithms

Let $X,Y$ be normal bounded operators on a Hilbert space such that $e^X=e^Y$. If the spectra of $X$ and $Y$ are contained in the strip $\s$ of the complex plane defined by $|\Im(z)|\leq \pi$, we show that $|X|=|Y|$. If $Y$ is only assumed to be bounded, then $|X|Y=Y|X|$. We give a formula for $X-Y$ in terms of spectral projections of $X$ and $Y$ provided that $X,Y$ are normal and $e^X=e^Y$. If $X$ is an unbounded self-adjoint operator, which does not have $(2k+1) \pi$, $k \in \ZZ$, as eigenvalues, and $Y$ is normal with spectrum in $\s$ satisfying $e^{iX}=e^Y$, then $Y \in \{\, e^{iX} \, \}"$. We give alternative proofs and generalizations of results on normal operator exponentials proved by Ch. Schmoeger.