  In this paper, we will establish a logarithmic weighted Adams inequality in a
logarithmic weighted second order Sobolev space in the whole set of
$\mathbb{R}^{N}$. Using this result, we delve into the analysis of a weighted
fourth-order equation in $\mathbb{R}^{N}$. We assume that the non-linearity of
the equation exhibits either critical or subcritical exponential growth,
consistent with the Adams-type inequalities previously established. By applying
the Mountain Pass Theorem, we demonstrate the existence of a weak solution to
this problem. The primary challenge lies in the lack of compactness in the
energy caused by the critical exponential growth of the non-linear term $f$.
