Connected Lie group 1-dimensional isomorphic to $\mathbb{R}$ or to $S^1$

Question

Let $G$ a connected Lie group of dimension 1. Show that \begin{align} G \cong \mathbb{R} \, \, \, \text{or} \, \, \, G \cong S^1 \end{align}

I tried to read and understand the topic Connected, one-dimensional Lie groups , but I have some problem to compute the kernel of the exponential map to see that $\ker(\exp) = \{ 0 \}$ or $\ker(\exp)= r\mathbb{Z}$ for some $r>0$.

Any suggestions? Thanks in advance!