This expands out to log (n!). You can see this because
T(n) = T(n - 1) + log n
= T(n - 2) + log (n - 1) + log n
= T(n - 3) + log (n - 2) + log (n - 1) + log n
= ...
= T(0) + log 1 + log 2 + ... + log (n - 1) + log n
= T(0) + log n!
The exact answer depends on what T(0) is, but this is Θ(log n!) for any fixed constant value of T(0).
A note - using Stirling's approximation, Θ(log n!) = Θ(n log n). That might help you relate this back to existing complexity classes.
Hope this helps!