To explain Logan's origin you have to establish a little of the world he lives in. Logan 5 lives in a domed city known as "a perfect world of total pleasure" where the people are suppose to be renewed at the age of 30 during an event called Carousel. What no ones knows is the people are actually killed and that the renewal is just a myth meant to make people comfortable. The whole thing is meant to maintain the population of the city.
The ages of the populace are kept tracked of by the main computer by a Palm Flower Crystal implanted in the right hand. The color changes from white to yellow to green to red. At the age of thirty the Palm Flower Crystal blinks red. and on the thirtieth birthday known as Last Day they go to renewal.
Some people are afraid that renewal is a myth and and attempt to run away at the age of 30. When this happens it is the job of a Sandman to find then and nullify (kill) them. That is where Logan 5 comes in. Logan 5 is a Sandman who fully believes in renewal. Things go fine until he nullifies a runner who carries on Ankh symbol on his.
When Logan if debriefed by the main computer he finds that the Ankh symbol is also the symbol for a place known as sanctuary where runners are trying to escape to. The computer then changes Logan 5's age from 26 to 30 to allow him to infiltrate the underground movement. Logan goes along with the plan even though he does not know if the computer will give him his four years back.
He meets up with a woman known as Jessica 6 whom he met earlier and who wears the Ankh around her neck. He tries to get her friends to show them where sanctuary is and at first they try to kill him but are eventually convinced that he is really trying to find sanctuary.
The two find out the sanctuary doesn't exist and all the others that went before them were killed when that were frozen by an automated robot that froze them as meat. As it had done for all the decades before. Not realizing the how the society had changed and that it was a forgotten relic of a past no one remembered.
When Logan and Jessica escape they find themselves in an overgrown and abandon city of Washington D. C. they also find an old wan who is like an alien to them as they have never seen someone grow older than 30. They also find that they were tracked by Francis 7 another Sandman and best friend of Logan who fights Logan to his own death. He was just unable to see that the reasons for their culture were no longer needed and that the Palm Flower Crystal didn't even work outside the city.
Logan and Jessica decide to go back to the dome city and show everyone what they have discovered. They even bring the old man with them but were unable to get him in the city. Once they are in the city they are not believed and are taken to the main computer that is unable to make sense of their discovery. So much so that the computer over loads and creates a breach on the dome where the people see the outside world for the first time and see the old man. Changing their presumptions forever.
Although the concept of Logan 5 was created in the novel Logan's Run the character is very different in the movie by the same name. He is really a different character. The Novel takes place in a different time and the maximum age is changed from 21 to 30. The city is also on another planet where Logan 5 in on Earth somewhere near Washington D.C. They even have different names where Logan 3 is in the novel and Logan 5 is in the novel.
The character from the movie is the one that is transferred to comic book form. The same characters were used in the television series although they were played by different actors. These and some comics continued the story from the film. The show however only lasted about fourteen episodes.