Implications of approximating the determinant
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).