The presence of nonlinear instabilities represents an important obstacle when it comes to turbulence simulation. In both compressible and incompressible flows, preserving kinetic energy has been shown to mitigate the stability issue that is due to the accumulation of the aliasing error. However, in the compressible case, errors on thermodynamic quantities can equally be the source of instabilities. Particular attention has been devoted to entropy preservation, following the second law of thermodynamics. Multiple discretizations of the compressible Euler equations have been proposed throughout the years in order to alleviate these problems. In this paper, we revisit these methods and propose new kinetic-energy-preserving and entropy-preserving formulations. In particular, the logic of the logarithmic mean used in finite element and finite volume methods has been extended to finite difference method to create new schemes capable of preserving both total energy and entropy. The different schemes have been compared in terms of preservation properties of global quantities, such as kinetic energy and entropy. In addition, we also include a computational cost analysis.