Statistical Inference — MLE of a log-scale shifted exponential parameter

Let $x_1, x_2, x_3, …, x_n$ be a random sample from the distribution having pdf $f(x, \theta) = \frac{1}{\theta}e^{-\frac{(x-\theta)}{\theta}}$; $x \ge \theta$, $\theta > 0$. The maximum likelihood estimator of $\ln\theta$ is