I've been having trouble with an assignment I received with the course I am following. The assignment in question:
Use induction to prove that when $n \geq 2$ is an exact power of $2$, the solution of the recurrence
T(n) = \begin{cases} 2 & \text{ if } n = 2,\\ 2T(n/2)+n & \text{ if } n =2^k, k > 1 \\ \end{cases}
is $T(n) = n\log(n)$
NOTE: the logarithms in the assignment have base $2$.
The base case here is obvious, when $n = 2$, we have that $2 = 2\log(2)$. However, I am stuck on the step here and I am not sure how to solve this.