I refer to the following theorem:
from 'Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations' by Werner Balser.
I can see that any matrices $M$ satisfying $\exp(2 \pi i M) = C$ whose eigenvalues have real parts contained in $[0,1)$ must be unique up to similarity, essentially through Jordan form considerations.
However I do not know how to go from this, or the result stated before the final uniqueness part of the theorem in the attached image, namely 'The eigenvalues of any two such $M$ can only differ by integers', which I do see must be true, to the statement that there is really a unique matrix $M$ whose eigenvalues are in the strip $0 \leq \text{Re}(z) < 1$ with $\exp(2 \pi i M) = C$.
Thank you in advance for any help.