We present an equation of motion coupled cluster approach for calculating and understanding intrinsic inelastic losses in core level x-ray absorption spectra (XAS). The method is based on a factorization of the transition amplitude in the time-domain, which leads to a convolution of an effective one-body spectrum and the core-hole spectral function. The spectral function characterizes these losses in terms of shake-up excitations and satellites, and is calculated using a cumulant representation of the core-hole Greens function that includes non-linear corrections. The one-body spectrum also includes orthogonality corrections that enhance the XAS at the edge.