Trouble understanding the master theorem, from Jeffrey Erickson's Notes
You can further simplify $n^{\log_b(k)}f(n)$.

We have $\frac{a}{k}f(\frac{n}{b}) = f(n)$. If $c = \log_b(n)$, we can apply this identity $c$ times to get $\left(\frac{a}{k}\right)^c f(1)$.

So $n^{\log_b(k)}f(n) = n^{\log_b(k)}\left(\frac{a}{k}\right)^c f(1) = k^{\log_b(n)}\left(\frac{a}{k}\right)^{\log_b(n)} f(1) = a^{\log_b(n)}f(1) = \Theta(n^{\log_b(a)})$