What message do George Clayton Johnson and William F. Nolan intend to communicate in Logan's Run?

The authors’ goal in writing Logan’s Run was to reveal the danger of authoritarianism. Authors George Clayton Johnson and William F. Nolan intend to show how unlimited power threatens ordinary citizens. These warnings are expressed through the lens of a dystopian society in which age-based discrimination and mandatory sexual activity are combined with strict limits on a person’s lifespan.

A closely related theme is the hypocrisy that can result when policies supposedly based on securing the common well-being of the majority of a society’s members are used to justify the repression of dissenting individuals. Along the same line, the authors show the importance of nonconformity and individuality, even when efforts at resistance seem futile.

The dangerously repressive role of the state is shown through the enforcement of Lastday—the day before one's mandatory euthanasia at the age of twenty-one. This practice is enforced by a predatory police force, the Sandmen, and inaccurately framed as voluntary.

The necessity and difficulty of resistance are portrayed through the complex actions of Logan and Jessica as they escape and seek Sanctuary. The character of Logan embodies the importance of individual free choice as Logan rejects his Sandman role and the authoritarianism it represents, in part through finding love as something beyond sexual gratification.

What are the key passages in George Clayton Johnson and William F. Nolan's Logan's Run?

I am working from an electronic text that doesn't have page numbers, so I will include chapters for your reference.

Don't get me wrong, citizen. I'm no coward. I'm not going to run. I have my pride. The system is right, I know that. World can only support so much life. Got to be a way to keep the population down…I've been loyal and I won't change now. (chapter 10)

Logan's society depends on an acceptance that the earth has surpassed capacity and is capable of sustaining only a set amount of human life. The populace has been brainwashed to believe that "volunteering" for death at age twenty-one is noble; those who don't show up "voluntarily" are hunted down and mercilessly executed. Still, Sawyer explains to Logan that he understands the necessity of his own death as a sacrifice for the survival of mankind.

Runners say please; runners say help; runners say mercy, runners say don't.
Doyle had said Sanctuary.
And Logan held a key which might lead to it, to a goal never proved to exist, to a place which could not exist. Not in this world. Not for a runner in 2116.
But what if Sanctuary were a reality? A place where runners were safe from the Gun. What if he, Logan 3, could find it and destroy it in the last twenty four hours of his life? His existence would be
justified; he'd be a world hero; his life would end in glory.
It would be a risk worth taking. And the key to the quest lay in his hand. (chapter 9)

When Logan faces his own Lastday, he is presented with a new challenge. Though he remains fully indoctrinated into the ideals of his community, Logan imagines using the final twenty-four hours of life to dismantle this "sanctuary" of individuals who seek to circumvent the expectations of their society.

"You're a monster. You've chased and killed people like my brother, whose only sin was wanting to live another day."
"I didn't kill your brother."
"Maybe not, but you would have. You'd have put a homer in him and been proud of yourself for doing it."
He had no answer. (chapter 8)

Jessica helps Logan to feel the weight of his own actions. She doesn't allow him to avoid personal responsibility for her brother's death because he has actively searched for and killed people just like her brother. These standards of "honor" and "sacrifice" are only upheld because people blindly accept lies as truth; Jessica helps dismantle Logan's unwavering belief in his training.

Logan stood there; feeling the sick emptiness flush through him. Why had she made him do this? Why hadn't she accepted Sleep? Why had she run?
…run
run…
Run!
And he was twenty-one. Suddenly, twenty-one! (chapter 3)

As he faces the likely possibility that he, too, will die according to the rules of his society, Logan recalls the stories of his own life. He has been awakened to the true brevity of twenty-one years of life, and he feels compelled to run towards longevity, abandoning the expectations of his world in an instinctual pursuit of survival.

To continue evaluating the text and determining what passages may be important for your purposes, you might try looking for quotes which expose themes of deception or betrayal. You might also find quotes which reflect significant moments of Logan's character development or his transformation from an antihero to a more traditional hero.