Constructing a DFA $M$ such that $L(M) = L(A) \bigtriangleup L(B)$ with a kind of log-space TM
Don't construct machines for $L(A)\cap L(B)$ and $L(A)\cup L(B)$. Use a variant of the product construction for those two automata to directly construct an automaton for symmetric difference. Because of the way the product construction works, you don't need to store much intermediate data at all: essentially the algorithm is just a couple of for loops and, with at most $n$&nbsp;states to loop through, you only need $\log n$ bits to hold the value of the loop counter.