Summary: In this note, we study the integrodifferential operator \((I - {\Delta})^{\log}\) corresponding to the logarithmic symbol \(\log(1 + | \xi |^2)\), which is a singular integral operator given by \[ (I - {\Delta})^{\log} u(x) = d_N \int_{\mathbb{R}^N} \frac{u (x) - u (x + y)}{ | y |^N} \omega(| y |) d y, \] where \(d_N = \pi^{- \frac{ N}{ 2}}, \omega(r) = 2^{1 - \frac{ N}{ 2}} r^{\frac{ N}{ 2}} K_{\frac{ N}{ 2}}(r)\) and \(K_\nu\) is the modified Bessel function of second kind with index \(\nu \). This operator is the Lévy generator of the variance gamma process and arises as derivative \(\partial_s |_{s = 0} ( I - {\Delta})^s\) of fractional relativistic Schrödinger operators at \(s = 0\). In order to study associated Dirichlet problems in bounded domains, we first introduce the functional analytic framework and some properties related to \(( I - {\Delta})^{\log} \), which allow to characterize the induced eigenvalue problem and Faber-Krahn type inequality. We also derive a decay estimate in \(\mathbb{R}^N\) of the Poisson problem and investigate small order asymptotics \(s \to 0^+\) of Dirichlet eigenvalues and eigenfunctions of \(( I - {\Delta})^s\) in a bounded open Lipschitz set.