The collective phenomena arising in a system of globally coupled chaotic logarithmic maps are investigated by considering 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 previously studied globally coupled systems, no splitting of the elements into different groups 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. The role that the properties of the local dynamics play in the emergence and characteristics of nontrivial collective behavior in globally coupled systems is discussed.