Fractional Sobolev spaces

Question

Suppose I have a function $f(x)=x^\alpha\log(x)^k$ defined on $(0,1)$ where $\alpha$ is complex and $k$ a non-negative integer. What is the sharpest result on which fractional Sobolev space $H^s((0,1))$ that $f$ belongs to? I expect that the result is $\mathrm{Re}(\alpha)+1/2-\epsilon$ for $\epsilon>0$ but am not really sure how to proceed...