How to prove $T(n) = 2T(n/2) + n/\log(n)$ can't be solved using the Master Theorem?
Your recursion can be written as:
$$
T(n) = aT(n/b) + f(n),
$$
where $a=b=2$ and $f(n) = n \log^{-1} n$. Defining $c_{crit} = \log_b a = 1$ you can see that $f(n) = \Theta( n^{c_{crit}} \log^k n)$ for $k=-1$, therefore case 2b of the Master Theorem applies.
The solution to the recurrence is therefore:
$$T(n) = \Theta( n^{c_{crit}} \log \log n ) = \Theta( n \log \log n).$$