Why was it that Light's father could see Light's lifespan, while Mikami couldn't see it (which allowed him to deduce that Light was Kira)? I was talking with a friend about Death Note and he said that he could't understand one thing:

When Light's father died, he saw Light's lifespan with his Shinigami eyes, so he could "confirm" that Light was not Kira. However, at the end of the story, Mikami looked at Light with his Shinigami eyes and recognized him, because he could not see his lifespan.

Is this actually a slight mistake in the story, or have I missed anything? (It's probably the second one)
In chapter 71, Light was setting up a plan that would prove himself to be innocent of being the killer (Kira); would give the NPA (National Police Agency of Japan) the advantage over Mello; and would keep him safe if Mello would ever get hold of his picture.

In order to kill Mello, he made preparations for a raid to take place at November 10 at 23:59. Almost everybody in Mello's organization would die, but in order to kill Mello himself he needed the help of the NPA. He sent them the Death Note together with Ryuku, making it possibly for one of the NPA members to bargain for the Death Eyes. He had expected Matsuda Touta to do the deal, since it went with his character, but in the end it was his father, Soichiro Yagami, who made the deal with Ryuku. With his father having the Death Eyes, it would be impossible for him to be owner of any Death Note, and therefore, he had relinquished ownership himself, instead of making Misa relinquish ownership over hers. Also he knew it would be dangerous to be the owner of any Death Note, since he knew Jack Neylon aka Kal Snyder had the Death Eyes and it would be as easy for Mello to kill him as it would be for Light to find and kill Kal Snyder. Of course, he was not going to stop being the killer (Kira) soon, so to retain his memories he strapped Misa's Death Note to his body. By doing so, he thought he would be able to kill Mello and prove himself innocent once again for the NPA. He failed to kill Mello through this raid, though.



In chapter 72, you can see that in addition to what I said above, Light also needed Ryuku to be there at the raid, to stop Sidoh from helping the "mafia". He knew Ryuku would be so kind as to harass Sidoh a bit ^^.



In chapter 74, you can see Light holding the Death Note when his father dies. After an owner dies, the ownership will go to the first person that picks up or touches the Death Note, and therefore Light becomes the owner again.



In chapter 81, Misa is ordered to relinquish her ownership to Teru Mikami. At this point there are two human Death Note owners and they are Light and Mikami, and Misa becomes an ordinary housewife. Therefore, Mikami is able to identify Light as the owner of the Death Note and the real "Killer".



For a summary of the Notes' owners the website provided by Maroon is really useful. See here.