# 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 $$\|\log u-\log v\| where $$C$$ is a constant independent of the dimension of the matrix. Thanks!