Calculating the runtime for a recursive algorithm
1) $f(n) \in O(n)$, since $k$ is a constant.

2) If $a=b=k$, then $\log_b a = \log_k k$. Your specific case of $\log_1 1$ only appears for $k=1$. And if we insert $k=1$ into the original recurrence we get $T(n) = T(n) + n + 1$, which does not make much sense as a description of a runtime. So it is sensible to simply exclude this case.