Upon Fourier transformation $x\mapsto k$, this becomes a diagonal operator with matrix elements $\langle k|\ln D|k'\rangle=2\pi \delta(k-k')\ln k$. So to find the matrix elements in the $x$-representation we would need to inverse the Fourier transform of the logarithm $\ln k$. From this [MSE answer](https://math.stackexchange.com/a/2353777/87355) for the Fourier transform of $\ln |k|$ (with absolute value signs) I would conclude that
$$\langle x|\ln D|x'\rangle=\bigl(\frac{i \pi}{2}-\gamma\bigr)  \delta (k)+\text{P.V.}\bigl(\frac{1}{2 k}-\frac{1}{2 | k| }\bigr).$$