Log-space transducer

A log space transducer (LST) is a type of Turing machine used for log-space reductions.
A log space transducer, formula_1, has three tapes:
formula_1 will be designed to compute a log-space computable function formula_4 (where formula_5 is the alphabet of both the "input" and "output" tapes). If formula_1 is executed with formula_7 on its "input" tape, when the machine halts, it will have formula_8 remaining on its "output" tape.
A language formula_9 is said to be log-space reducible to a language formula_10 if there exists a log-space computable function, formula_11, which will convert an input from problem formula_12 into an input to problem formula_13. I.E. formula_14.
This seems like a rather convoluted idea, but it has two useful properties that are desirable for a reduction:
Transitivity holds because it is possible to feed the output tape of one reducer (A→B) to another (B→C). At first glance, this seems incorrect because the A→C reducer needs to store the output tape from the A→B reducer onto the work tape in order to feed it into the B→C reducer, but this is not true. Each time the B→C reducer needs to access its input tape, the A→C reducer can re-run the A→B reducer, and so the output of the A→B reducer never needs to be stored entirely at once.