We describe a single-channel speech enhancement algorithm that is based on modulation-domain Kalman filtering that tracks the inter-frame time evolution of the speech logpower spectrum in combination with the long-term average speech log-spectrum. We use offline-trained log-power spectrum global priors incorporated in the Kalman filter prediction and update steps for enhancing noise suppression. In particular, we train and utilize Gaussian mixture model priors for speech in the log-spectral domain that are normalized with respect to the active speech level. The Kalman filter update step uses the log-power spectrum global priors together with the local priors obtained from the Kalman filter prediction step. The logspectrum Kalman filtering algorithm, which uses the theoretical phase factor distribution and improves the modeling of the modulation features, is evaluated in terms of speech quality. Different algorithm configurations, dependent on whether global priors and/or Kalman filter noise tracking are used, are compared in various noise types.