The Riemann surface of the logarithm is the complex plane over the punctured plane via exp

For each $w\in \mathbb{C}$, let $V_w:=e^{w}B(1,1\mathrm{/}2)$ and define ${\mathrm{\ell }}_w(u):=w+\mathrm{Log}(e^{-w}u)$ for $u\in V_w$. Because $B(1,1\mathrm{/}2)\subseteq S$, fact [L1] makes ${\mathrm{\ell }}_w$ holomorphic on $V_w$, with $\exp ({\mathrm{\ell }}_w(u))=e^w{e}^{\mathrm{Log}(e^{-w}u)}=u$ and ${\mathrm{\ell }}_w(e^w)=w$. Thus $\mathrm{\Phi }(w):=[{\mathrm{\ell }}_w{]}_{e^w}$ is a well-defined logarithm germ over $e^w$.

The germ $\mathrm{\Phi }(w)$ lies on ${\mathcal{R}}_{\log }$. Indeed, along the path ${\gamma }_w(t)=e^{tw}$ from $1$ to $e^w$, choose a subdivision fine enough that for every adjacent pair $t_j,t_{j+1}$ and every $t\in [t_j,t_{j+1}]$ one has $\mathrm{\mid }{e}^{(t-t_j)w}-1\mathrm{\mid }<\frac{1}{2},$ and also $\mathrm{\mid }\mathrm{Im}((t_{j+1}-t_j)w)\mathrm{\mid }<\pi$. The first condition puts the entire subpath ${\gamma }_w([t_j,t_{j+1}])$ inside $V_{t_{j}w}$. At the joining point, the second condition gives ${\mathrm{\ell }}_{t_{j}w}(e^{t_{j+1}w})=t_{j}w+\mathrm{Log}(e^{(t_{j+1}-t_j)w})=t_{j+1}w,$ so consecutive branches agree there as germs. Starting at $t_0=0$ gives the principal logarithm germ at $1$, and the final germ is $\mathrm{\Phi }(w)$.

If $\mathrm{\Phi }(w)=\mathrm{\Phi }(v)$, then the two germs have the same base point and the same value at that base point. Therefore $e^w=e^v$ and $w=v$. So $\mathrm{\Phi }$ is injective.

Let $\xi =[\mathrm{\ell }{]}_z$ be any point of ${\mathcal{R}}_{\log }$ and put $w:=\mathrm{\ell }(z)$. Shrink the representative domain of $\mathrm{\ell }$ to a disc $U\subseteq V_w$ on which $\mathrm{\ell }(U)\subseteq B(w,\pi )$. On $U$ one has $\exp (\mathrm{\ell }(u))=u=\exp ({\mathrm{\ell }}_w(u))$, so [L2] gives $\mathrm{\ell }(u)-{\mathrm{\ell }}_w(u)\in 2\pi i\mathbb{Z}$ for every $u\in U$. The difference is continuous, takes the value $0$ at $z$, and its image lies in the discrete set $2\pi i\mathbb{Z}$; because $U$ is connected, the difference is identically $0$. Hence $\xi =\mathrm{\Phi }(w)$, and $\mathrm{\Phi }$ is surjective.

The map $\mathrm{\Lambda }$ is the inverse of $\mathrm{\Phi }$ by steps 2.1 and 2.2. In a chart $N(\mathrm{\ell },U)$ on the germ surface one has $\mathrm{\Lambda }\circ {\varphi }_{\mathrm{\ell },U}^{-1}(u)=\mathrm{\ell }(u)$, so [L3] makes $\mathrm{\Lambda }$ holomorphic. Near any $w_0\in \mathbb{C}$, if $\mathrm{\mid }w-w_0\mathrm{\mid }<1\mathrm{/}4$ then $\mathrm{\Phi }(w)\in N({\mathrm{\ell }}_{w_0},V_{w_0})$ and ${\varphi }_{{\mathrm{\ell }}_{w_0},V_{w_0}}(\mathrm{\Phi }(w))=e^w$, which is holomorphic in $w$; so $\mathrm{\Phi }$ is holomorphic as well. Therefore $\mathrm{\Lambda }$ is a biholomorphism.

For every $[\mathrm{\ell }{]}_z\in {\mathcal{R}}_{\log }$ one has $p([\mathrm{\ell }{]}_z)=z=\exp (\mathrm{\ell }(z))=\exp (\mathrm{\Lambda }([\mathrm{\ell }{]}_z))$, so $p=\exp \circ \mathrm{\Lambda }$.