Implications of approximating the determinant It is known that one can compute exactly the determinant of an $n\times n$ matrix in determinstic $\log^2(n)$ space.
What would be the complexity implications of approximating the determinant of a real matrix, of norm at most $1$ ($\left\|A\right\|\leq 1$) in randomized logarithmic space, up to say, a $1/\text{poly}$ accuracy?

In this respect, what would be the "correct" approximation to ask for - multiplicative or additive?  (see one of the answers below).
With a risk of not having understood the details of the question properly: Being able to approximate the determinant within any factor requires being able to decide whether a square matrix is singular or not, which should have some consequences.

For one thing, it gives a randomized test for whether a general graph has a perfect matching (via the Tutte matrix and Schwarz-Zippel). I don't think the latter is known in randomized logspace (e.g., the Complexity Zoo lists bipartite perfect matching as hard for NL).