This problem was done in class during the first week of November. The only thing missing was the Expected value of our max likely estimator lambda. Since we know:
$ \lambda_{ML}=\frac{1}{x}\, $
we know
$ E[\lambda_{ML}]=\frac{1}{E[x]}\, $
and the expected value of an exponential random variable is $ \frac{1}{\lambda}\, $
Thus our estimator is unbiased
