The problem is a simple syntax, code typing, issue. It is Log, not log. Also, if you want the plot to change between evaluations you should make the definition of randomFunc to be a delayed assignment.
proc = ItoProcess[
\[DifferentialD]x[t] == 0.1 x[t] - 0.3 Log[x[t]] \[DifferentialD]t + 0.1*x[t] \[DifferentialD]w[t],
x[t],
{x, 1}, t,
w \[Distributed] WienerProcess[]
]
randomFunc := RandomFunction[proc, {0., 5., 0.01}]
ListLinePlot[randomFunc, Filling -> Axis]
