Question about injectivity of exponential mapping between SE(3) and se(3)

Question

If we denote $X, Y \in se(3)$, and they have this relationship

$$e^X = e^Y$$

is it safe to assume that $X = Y$ for every element? If it is not, may I know the case when it is not?

Intuitively, the rotation mapping is not injective because if an object is rotated around certain axis for $\theta$ and $\theta + 2\pi$, they will result in the same pose. But is there another case?

Thanks.

Answer 1

Agree with Deitrich if $||x||\in\mathbb{R}^+$, when the magnitude of exponential coordinates within $\pi$, the exponential mapping is surjective so it will be safe.