Base c making a Poisson exponential a martingale

Let $N(t)$ be a Poisson process of rate $\lambda>0$. For a constant $c>0$ define $S(t)=\exp\!\big(N(t)\log c-\lambda t\big)=c^{N(t)}e^{-\lambda t}$. For which value of $c$ is $S(t)$ a martingale?

$c=2$. By independent increments, $\mathbb{E}[S_t\mid\mathcal F_s]=S_s\,e^{-\lambda(t-s)}\,\mathbb{E}[c^{N(t)-N(s)}]=S_s\,e^{\lambda(t-s)(c-2)}$, using the Poisson generating function $\mathbb{E}[c^{N}]=e^{\mu(c-1)}$. This equals $S_s$ for all $s<t$ only when $c=2$.