Has any one seen function of the following type? Define

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

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!