In an earlier thread, I asked for help finding the UMVUE for e^-(t-theta). Now, I am looking to find the maximum likelihood estimator for the same density function (where theta <= t < infinity).

I got as far as getting L(theta) = e^-(sum from 1 to n t)*e^n*theta. From here, it feels like I should take the log of this and then take the derivative and then solve for 0 but I am not sure if I am doing it right. Any help?

You're missing a very important piece of the likelihood and what you have right now should give some indication of what you're missing. Right now the piece of your likelihood that depends on the data ( e^(sum of the tis) ) doesn't depend on theta at all. So if we wanted to maximize the likelihood with respect to theta we can ignore it. e^(n*theta) grows with respect to theta so clearly to maximize this you just want to make theta as large as possible. With no bounds on what theta can be in your likelihood directly the MLE doesn't exist.

But is there a piece of information that you could use that would put a bound on what theta could be?