Let $X_1, \dotsc, X_t \overset{d}{=} X$, i.i.d., with $E[X]=\lambda<1$ and define $I:=\lambda - 1 - \log \lambda$.
We want to show that $P[X_1 + \dotsb + X_t \geq t ] \leq e^{-I t}$ using the Chernoff bound (exponential Markov inequality)
To do this, we rewrite for some $s>0$
$$P[X_1 + \dotsb + X_t \geq t] = P[e^{s (X_1 + \dotsb + X_t)} \geq e^{st}]\leq \frac{E[e^{s (X_1 + \dotsb + X_t)}]}{e^{st}}= \frac{(E[e^{sX}])^t}{e^{st}}$$ $$=\exp(t (log(E[e^{sX}])-s))$$
but I have no idea how one gets the $I$ out of it, so in particular which $s$ to choose. Do you have some hints for me?