A family of positivity-preserving lattice Boltzmann methods (LBMs) is proposed for compressible flow simulations in the continuum regime. It relies on the efficient collide-and-stream algorithm with a collision step based on exponential distribution functions. The latter serves as a generalization of Grad's post-collision distribution functions for which here (1) the linearized non-equilibrium contributions are replaced by their exponential forms and (2) the number of these contributions can be chosen arbitrary. In practice, post-collision moments of our exponential formulation are enforced through an iterative moment-matching approach to recover any macroscopic physics of interest, with or without external forces. This methodology directly flows from the extended framework on numerical equilibria [J. Latt et al., Philos. Trans. R. Soc. A 378, 20190559 (2020)] and goes one step further by allowing for the independent relaxation of hydrodynamic and high-order modes in a given moment space, notably, making the Prandtl number freely adjustable. The model is supplemented by a shock-capturing technique, based on the deviation of non-equilibrium moments from their equilibrium counterparts, to ensure good numerical properties of the model in inviscid and under-resolved conditions. A second exponential distribution accounts for extra degrees of freedom of molecules and allows for the simulation of polyatomic gases. To validate this novel approach and to quantify the accuracy of different lattices and moment closures, several 2D benchmark tests of increasing complexity are considered: double shear layer, linear wave decay, Poiseuille flow, Riemann problem, compressible Blasius flow over a flat plate, and supersonic flow past an airfoil. Corresponding results confirm the accuracy and stability properties of our approach for the simulation of compressible flows with LBMs. Eventually, the performance analysis further highlights its efficiency on general purpose graphical processing units.