Induction for recurrence: $t(n) = t(n/2) + \log_2 (n)$

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Assume $t(1) = 0$. Assume $n$ is a power of $2$, so $n = 2^m$ or $m = \log_2 n$.

I am first asked to solve the recurrence, then asked to prove my answer by induction.

I solved the recurrence and got $O((\log_2(n)^2)$, but dont understand the second part.

Any help is appreciated.