The use of a logarithmic misfit function has proven to be advantageous for full waveform inversion of field data in noisy environments. However, few efforts have been carried out to define an efficient and effective updating procedure. In this paper, we utilize an approximated Gauss-Newton method, which is called CGLS or Gauss-Newton-CG method, using a logarithmic misfit for the frequency-domain full waveform inversion. The Gauss-Newton inversion via a conjugate gradient algorithm is an efficient method because the Hessian matrix does not need to be explicitly calculated. In this method, the logarithmic objective function is not applied even though it is robust against noise and able to resolve strongly damped wavefields such as the Laplace-domain wavefields. In this context, we use the logarithmic objective function to formulate the normal equation for implementation in the Gauss-Newton inversion. This results in an unstable inverse problem, which is mitigated by applying a heuristic filtering method to the gradient direction and the Hessian matrix. A simple damping term was added to the approximated Hessian matrix to stabilize the inversion. To verify our algorithm, we compare the proposed Gauss-Newton method with the gradient method. The Marmousi model and the SEG/EAGE salt model were used for this comparison. The gradient method produces distorted images in the deeper parts of the velocity model, while our method gives improved results and shows noticeable improvement for recovering the structure beneath the salt tayer in the SEG/EAGE model.