linear algebra - Matrix Logarithms are Not Unique

In my ODE class, we proved that if exp(L) = exp(L') then the eigenvalues are congruent mod 2πi. Here L, L' are two nxn matrices. I wanted to know if something more precise was true.

In a way, we should expect that matrix logs are multiple valued since this is the case in $mathbb{C}$. log(re^iθ) = log r + iθ + 2πik with $k in mathbb{Z}$. In this way we can construct an branched infinite cover of the complex plane.

We'll define multiplcity mod 2πi of an eigenvalue λ to be the number of eigenvalues congruent to λ mod 2πi up to multiplicity. If exp(L) = exp(L') are the spectra of L and L' the same including multiplicity mod 2πi?

To put this another way, I could imagine two 5x5 matrices exp(L) = exp(L') where

• the spectrum of L is (λ₁, λ₁, λ₂, λ₂ + 2πi, λ₂ + 4πi)

while

• the spectrum of L' is (λ₁, λ₁, λ₁, λ₂ + 2πi, λ₂ + 4πi)

Here the multiplicities mod 2πi are different. One would be (2,3) while the other would be (3,2). Could exp(L) = exp(L') in this case?