From Statistical Inference by Casella and Berger:
Let $X_1, \dots, X_n$ be a random sample from an exponential population with pdf $f(x| \theta) = e^{-(x-\theta)}$ if $x \ge \theta$ and $0$ otherwise.
Consider testing $H_0 : \theta \le \theta_0$ verses $H_1: \theta \gt \theta_0$. $L(\theta\mid x)$ is an increasing function of $\theta$ on $-\infty \lt \theta \le x_{(1)}$
Then the likelihood ratio test statistic is $\lambda(x) = \{1 \text{ if } x_{(1)} \le \theta_0 \text{ and } e^{-n(x_{(1)} - \theta_0)} \text{ if $x_{(1)} \gt \theta_0\}$ }$
A likelihood ratio test (one that rejects $H_0$ if $\lambda(X) \le c$) is a test with rejection region $\{x : x_{(1)} \le \theta_0 - \frac{\log c}{n}\}$
Why is the rejection region just $\{x : x_{(1)} \le \theta_0 - \frac{\log c}{n}\}$? Why is the author ignoring the other part of $\lambda(x)$?