Uniform convergence of logarithm on a compact set.

Question

Given that $\phi$ and $\phi_n,n \geq 1$ are continuous functions from $\mathbb{R}$ into $\mathbb{C}$ such that $ \phi(0) = \phi_n(0) = 1$, $\phi(x) \neq 0$ and $\phi_n(x) \neq 0$ for any $x$. Suppose that for any compact subset $K$ of $\mathbb{R}$ $$ \lim_{n\to \infty} \sup_{x \in K} \vert \phi(x) - \phi_n(x) \vert = 0. $$ Does this imply that $$ \forall x \in \mathbb{R}: \lim_{n \to \infty} \log \phi_n(x) = \log \phi(x)$$?