Why L is defined as L = SPACE$( \log n)$ instead of L = SPACE$(\log^2 n)$ or L = SPACE$(\sqrt n)$?
The complexity class $\mathsf{L}$ satisfies many desirable properties:


It is closed under concatenation and iteration.
The corresponding function class is closed under composition.
The same complexity class is obtained for any number of work tapes.
It is resilient under "reasonable" input transformations with polynomial blowup (see below).
It can accommodate most known NP-hardness reductions.
It is a subset of $\mathsf{P}$.
It supports pointers indexing the input.


What is an input transformation? Consider the case of graphs. We can encode a graph either as an adjacency matrix of as adjacency lists. We can convert between the two in logspace, and so a graph problem which is in $\mathsf{L}$ under one of them is also in $\mathsf{L}$ under the other.

Logspace is the natural space analog of $\mathsf{P}$, considering the containment $\mathsf{SPACE}(f(n)) \subseteq \mathsf{TIME}(2^{f(n)})$.

It also shows up in the refined Schaefer's dichotomy theorem, as the lowest non-trivial complexity class.