Has anyone seen any function of the following type?

$$ g(x):=\sum_{n=0}^\infty \frac{x^n}{n!}\exp\left(-\frac{a^n}{x}\right),\quad a>1,x\ge 0. $$

The question is whether for some constant $c>0$,

$$ \lim_{x\rightarrow\infty}\frac{1}{x}\log g(x) \ge c. $$

Thanks a lot for any hints!