Why is $|\log(\|x\|)|^k$ in $H^1(\Omega)$ when $0<k<1/2$ and $\Omega$ is the $2D$ unit ball?

Question

Let $\Omega= B_1(0)$ the unit ball in $\mathbb{R^2}$. How to prove that for $0<k<1/2$ the function $v(x)= |\log(\|x\|)|^k$ is in

$$H^1(\Omega)=\left\{ f\in L^2(\Omega) : \frac{\partial f}{\partial x_i} \in L^2(\Omega) , i=1,2\right\}?$$