For every polynomial F in two complex variables we define the Łojasiewicz exponents ${ℒ}_{p,t}\left(F\right)$ measuring the growth of the gradient ∇F on the branches centered at points p at infinity such that F approaches t along γ. We calculate the exponents ${ℒ}_{p,t}\left(F\right)$ in terms of the local invariants of singularities of the pencil of projective curves associated with F.