  Let $ \{L_n\}_{n\geq 0} $ be the sequence of Lucas numbers. In this paper, we
look at the exponential Diophantine equation $L_n-2^x3^y=c$, for $n,x,y\in
\mathbb{Z}_{\ge0}$. We treat the cases $c\in -\mathbb{N}$, $c=0$ and $c\in
\mathbb{N}$ independently. In the cases that $c\in \mathbb{N}$ and $c\in
-\mathbb{N}$, we find all integers $c$ such that the Diophantine equation has
at least three solutions. These cases are treated independently since we employ
quite different techniques in proving the two cases.
