The phenomena of synchronization and nontrivial collective behavior in a system of globally coupled chaotic logarithmic maps are investigated through the properties of the mean field of the network. Several collective states are found in the phase diagram of the system: synchronized, collective periodic, collective chaotic, and fully turbulent states. In contrast with other globally coupled systems, no clustering nor quasiperiodic collective states occur in this model. The organization of the observed nontrivial collective states is related to the presence of unstable periodic orbits in the local dynamics.