This note is concerned with proving convergence rates for nonlinear Tikhonov regularization under mild regularity assumptions on the solution, namely source conditions of logarithmic type. For the choice of the regularization parameter, either a priori or a posteriori parameter choice strategies according to the discrepancy principle can be used. As usual in showing convergence rates for nonlinear ill-posed problems, restrictions on the nonlinearity of the forward operator have to be made unless the initial error is sufficiently smooth.