ABSTRACT. We show that LOGSPACE is characterised by finite orthogonal tail-recursive cons-free constructor term rewriting systems. This result is non-trivial, because in contrast to previous work on characterising LOGSPACE by tail-recursive cons-free programs we do not impose any fixed evaluation strategy. We provide a LOGSPACE algorithm which computes constructor normal forms. We then use this algorithm in the proof of our main result: that simple stream term rewriting systems characterise LOGSPACE-computable stream functions as defined by Ramyaa and Leivant. This result concerns characterising logarithmic-space computation on infinite streams by means of infinitary rewriting.