Define $ \mathcal C_R = \bracs{f \in C(\ol{B(0, R)}, \complex): f(0) = 1, f(\xi) \ne 0 \forall \xi \in \ol{B(0, R)}} $ and let $f \in \mathcal C_R$, then there exists a unique function $\ell_f \in \mathcal C_R$ such that $\exp \circ \ell_f = f$, known as the **principal logarithm** of $f$. If $\abs{f - 1} < 1/2$, then $\ell_f$ is the composition of $f$ and the principal logarithm, where $ \text{Log}(x) = \ln \abs{x} + i\theta \quad \theta \in (-\pi, \pi] $ If $\seq{f_n} \subset \cc_R$ such that $f_n \to f \in \cc_R$ uniformly, then $\ell_{f_n} \to \ell_{f}$ uniformly. Let $z \in \complex$ such that $\abs{z - 1} \le 1/2$, then $\abs{\log(z)} \le 2$. Let $z \in \complex$ such that $\re{z} \le 0$, then $\abs{1 - e^z} \le \abs{z}$.