We construct non-malleable extractors with seed length d = O(\log{n}+\log^{3}(1/\epsilon)) for n-bit sources with min-entropy k = \Omega(d), where \epsilon is the error guarantee. In particular, the seed length is logarithmic in n for \epsilon> 2^{-(\log{n})^{1/3}}. This improves upon existing constructions that either require super-logarithmic seed length even for constant error guarantee, or otherwise only support min-entropy n/\log^{O(1)}{n}.