In this paper we study languages accepted by nondeterministic $\log n$-tape automata that scan their input only once and relate their computational power to two-way $\log n$-tape automata. We show that for the one-way $\log n$-tape automata the nondeterministic model (1-NL) is computationally much more powerful than the deterministic model (1-L), that under one-way $\log n$-tape reductions there exist natural complete languages for these automata and that the complete languages cannot be sparse. Furthermore, we show that any language complete for nondeterministic one-way $\log n$-tape automata (under 1-L reductions) is also complete for the computationally more powerful nondeterministic two-way $\log n$-tape automata (NL) under two-way $\log n$-tape reductions. Therefore, for all bounds $T(n)$, $T(n) \geqq \log n$, the deterministic and nondeterministic $T(n)$-tape bounded computations collapse iff the nondeterministic one-way $\log n$-tape computations can be carried out by two-way deterministic log $n$-tape automata.