Decay estimate of Fourier transform with a log function

Question

Assume $f(x)$ ($x \in \mathbb{R}$) is a smooth function with a compact support. Assume $0 < \epsilon < 1$ and $i = \sqrt{-1}$ is the imaginary unit. Can we show that there exist a constant $C$ which is not dependent on $\epsilon$ such that the following estimate holds for $\xi \in \mathbb{R}$? \begin{equation} |\int_\mathbb{R} e^{-ix\xi}f(x) \text{log}(x + i\epsilon)dx| \le \frac{C}{1 + |\xi|} \end{equation}