recursion $t(n)=\sqrt{2} \times \frac{tn}{2} +\log{n}$

Question

I tried substituting $m=\log{n}$

$t(2^n)=\sqrt{2} \times \frac{t2^n}{2}+m t(m)= \sqrt(2) \times \frac{tm}{2}+m$

From here I got $\log {n}$ But with induction I proofed its $\sqrt {n}$