What condition is need for exponential equation to have unique solution?

Question

Goal: what condition we need so that there is a unique solution of $A \in \mathbb{C}^{N \times N}$

$$e^{A} = e^{\Lambda}$$ where $\Lambda \in \mathbb{\Lambda}^{N\times N}$ is known, it is complex diagonal matrix.

Answer 1

It's not unique. For all $z\in\mathbb{C}$ and $k\in\mathbb{Z}$ we have $e^z=e^{z+2\pi ik}$. From here it easily follows that the solution $A=\Lambda$ is not unique. For example, if $\Lambda$ is diagonal then you can define $B$ by adding $2\pi i$ to every element on the main diagonal and you will get $e^B=e^{\Lambda}$.