Let's consider a matrix $X \in M\left(2, \mathbb{R}\right)$. It's a well-known result that if $\lambda$ is an eigenvalue of $X$, then $\exp\left(\lambda\right)$ is an eigenvalue of $\exp\left(X \right)$.

But does it hold the other way round? More precisely, my question is as follows: is we know eigenvalues of $\exp\left(X\right)$, then what can we say about eigenvalues of $X$? Does existence of eigenvalues of $\exp\left(X\right)$ even guarantee existence of (real) eigenvalues of $X$?