Proof of big theta using induction
Hint: If $T(n/2) \leq M \frac{n}{2} \log \frac{n}{2}$ then
$$ T(n) = 2T(n/2) + cn \leq Mn\log \frac{n}{2} + cn \leq Mn\log n +n (c-M\log 2). $$
The lower bound should be similar.