  In this paper, we study the logarithmic Laplacian operator $L_\Delta$, which
is a singular integral operator with symbol $2\log |\zeta|$. We show that this
operator has the integral representation $$L_\Delta u(x) = c_{N}
\int_{\mathbb{R}^N } \frac{ u(x)1_{B_1(x)}(y)-u(y)}{|x-y|^{N} } dy + \rho_N
u(x) $$ with $c_N = \pi^{- \frac{N}{2}} \Gamma(\frac{N}{2})$ and $\rho_N=2 \log
2 + \psi(\frac{N}{2}) -\gamma$, where $\Gamma$ is the Gamma function, $\psi =
\frac{\Gamma'}{\Gamma}$ is the Digamma function and $\gamma= -\Gamma'(1)$ is
the Euler Mascheroni constant. This operator arises as formal derivative
$\partial_s \Big|_{s=0} (-\Delta)^s$ of fractional Laplacians at $s= 0$. We
develop the functional analytic framework for Dirichlet problems involving the
logarithmic Laplacian on bounded domains and use it to characterize the
asymptotics of principal Dirichlet eigenvalues and eigenfunctions of
$(-\Delta)^s$ as $s \to 0$. As a byproduct, we then derive a Faber-Krahn type
inequality for the principal Dirichlet eigenvalue of $L_\Delta$. Using this
inequality, we also establish conditions on domains giving rise to the maximum
principle in weak and strong forms. This allows us to also derive regularity up
to the boundary of solutions to corresponding Poisson problems.
