The author studies nonlinear equations of the form
is Fréchet-differentiable between Hilbert spaces
fails to be boundedly invertible at the solution
. This situation characterizes nonlinear illposed problems and arises in the study of inverse potential and inverse scattering problems. In this paper, logarithmic convergence rates of the iteratively regularized Gauß-Newton method are proven under a relatively weak source condition for the solution. For the inverse potential and the inverse scattering problem and the case of the obstacle being a circle, the author interpretes this condition as a smoothness assumption. A few numerical experiments show the applicability of the method with the expected convergence rates.