  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.
