Why is log(n) a space-constructible function?
Because when we write $\log n$ it's either in a context where it doesn't matter (e.g. Landau bounds) or with the implicit meaning of $\lfloor \log n \rfloor$ or $\lceil \log n \rceil$.

It certainly doesn't make sense to talk about constructible functions with real range, I would assume your source is talking about $\lfloor \log n \rfloor$ or $\lceil \log n \rceil$.