Q:

Why log-space reduction is used for NL-completeness while PSPACE reduction isn't used for PSPACE completeness?

NL-Complete languages are defined by Log-space reduction, while PSPACE complete languages are defined by poly-time many-to-one reduction.
According to these posts :
Why not polynomial-space reductions for $PSPACE$-hardness?
PSpace-completeness under PSpace reductions
Every PSPACE language would be PSPACE-Complete if we defined completeness using a
 Poly-SPACE reduction (instead of a poly-TIME reduction).
My question is, why does Log-space reduction doesn't imply completeness for every $L \in NL$, for the same reasoning?
Take any A,B $\in$ NL , and fixed $y,n$ s.t $y\in A$ and $n\notin A$.
We can define the following log-space reduction $f$ :
$$f(x)=\begin{cases}\ y&\text{if }w\in A\\
                    \ n&\text{if }w\notin A.\end{cases}$$
Just solve A in log-space and let our output be the fixed instance according to the right case.
How come log-space reductions are NOT useless for NL completeness while Pspace reductions are useless for PSPACE completeness? What am I missing ?

A:

Your reduction $f$ works in nondeterministic logspace, which is conjectured to be stronger than logspace. Assuming this conjecture, it follows that the concept of NL-completeness is not trivial, that is, not all problems in NL are NL-complete; in particular, problems in L are not NL-complete.
What might be confusing you is that PSPACE=NPSPACE, which is proved using Savitch's theorem. In the context of logarithmic space, the theorem only shows that problems in NL can be decided in space $O(\log^2 n)$.

