Explanation of a specific recurrence with respect to Master Theorem Concerning the Master Theorem. I have found the following equation as the base of analysis:  

$\quad T(n) = aT(n/b) + \Theta(n^k)$  

but I also found the following:  

$\quad T(n) = aT(n/b) + \Theta(n^k\cdot\log_p n)$  

where the base $p$ is a real number.  

Can anyone explain the second equation? I understand the proof with the first equation but can not understand the second formula.
Sometimes the master theorem is only given for recursions of the form $T(n) = aT(n/b) + \Theta(n^k)$, but the Wikipedia article includes a more general version which can handle functions of the form $T(n) = aT(n/b) + \Theta(n^k(\log n)^l)$, and even more general ones when $k \neq \log_b a$. The similar Akra-Bazzi theorem handles more general situations.