In Conrad and Brinon's notes http://math.stanford.edu/~conrad/papers/notes.pdf, two uniformizers of $B_{dR}$ are produced: one is $\xi := [\tilde{p}]-p$ (bottom of p.58), where $\tilde{p} = (p, p^{1/p}, p^{1/p^2}, \ldots)$ is some choice of compatible $p^r$ roots of unity, and the other is $t := \log(\epsilon)$, where $\epsilon$ is some basis of the Tate module of $\mathbb{C}_K$.

Furthermore, we know that $B_{dR}^+ / (t) \simeq \mathbb{C}_K$. Therefore, there should be some $a_1 \in \mathbb{C}_K$ such that $\xi = a_1t + ...$. My question is: how can we describe more explicitly what this $a_1$ is?