Bound on difference of log of unitary matrices

Suppose I have two unitary matrices $ u, v$ such that $ \|u-v\|<\epsilon$ in the operator norm. Is there a way to bound the quantity $ \|\log u-\log v\|$ ? We can assume that $ \epsilon$ is sufficiently small and we choose a branch cut for the logarithm such that eigenvalues of $ u$ and $ v$ will not be split up. Ideally I would like to get a bound in the form $ \|\log u-\log v\|<C\epsilon$ where $ C$ is a constant independent of the dimension of the matrix. Thanks!