but unfortunately it will not give the explicit answer for the general
case of n - it returns Log^(n)(x) in stead of the actual expression in
just n and x..
Ok, found a way. Use Lichtblau's method as shown here.
nthDeriv[f_, x_, n_] := n!*SeriesCoefficient[f[x], {x, x, n}]
And now type
nthDeriv[Log, x, n]
gives