Logspace complexity of functions and structures is based on the notion of a Turing machine with input and output as in Papadmitriou [16]. For any k > 1, we construct a logspace isomorphism between {0,1}^* and {0,1,..., k}^*. We improve results of Cenzer and Remmel [5] by characterizing the sets which are logspace isomorphic to {1}^*. We generalize Proposition 8.2 of [16] by giving upper bounds on the space complexity of compositions and use this to obtain the complexity of isomorphic copies of structures with different universes. Finally, we construct logspace models with standard universe {0,1}^* of various additive groups, including Z(p ^∞) and the rationals.