By the Cauchy Condensation Test, which is probably what you mean by a dyadic test, a decreasing series of positive terms $\sum_{n=2}^\infty f(n)$ converges if and only if $\sum_{k=2}^\infty 2^k f(2^k)$ converges.
In our case, $f(n)=\frac{1}{n(\ln n)^p}$, and therefore $$\sum_{k=2}^\infty 2^k f(2^k)=\sum_{k=2}^\infty \frac{1}{(\ln 2)^pk^p}.$$
Now we can use the fact that $\sum_2^\infty \frac{1}{k^p}$ converges if and only if $p\gt 1$.
**Remark:** If you want to prove that $\sum_{k=2}^\infty \frac{1}{k^p}$ converges iff $p\gt 1$, you can use Cauchy Condensation once more. One ends up with a geometric series.