Help! A question of MLE

There is no theta in that density. :) Be careful with what you write!

(But is it not pretty obvious? What happens if you take the log?)
I wrongly wrote lamda in the place of theta, a little error. But the question is that I don't know how to tackle with the absolute value of (x-θ) after taking the log and summing up


Ambassador to the humans
Can you show us what you have so far.

And completely unrelated... What properties do you know about the sample median?
Have you been tasked with or at least told that the median minimizes the mean absolute deviation?
The first time I heard such a conclusion,but it is a good hint that we only need to prove that the minimum of the the mean absolute deviation is the median. Case closed then