Recognizing the regime of positive definiteness for a strictly logarithmic covariance kernel, we prove that the small deviations of a related Gaussian multiplicative chaos (GMC) $M_γ$ are for each natural dimension $d$ always of lognormal type, i.e. the upper and lower limits as $t\to \infty$ of $$ -\ln\Big(\mathbb{P}(M_γ(B(0,r))\le δ\Big)/(\ln δ)^2 $$ are finite and bounded away from zero. We then place the small deviations in the context of Laplace transforms of $M_γ$ and discuss the explicit bounds on the associated constants. We also provide some new representations of the Laplace transform of GMC related to a strictly logarithmic covariance kernel.