Mikami had heard about Takada being kidnapped, so he took a guess as to what Kira would want him to do. So he wrote Takada's name down.
The thing was, Light also wrote her name down a few seconds later, but specified a cause of death, so the two name-writings both worked -- Mikami's for a time of death, Light's for a cause.
Gevanni, who had been tailing Mikami, noticed that he suddenly deviated from his obsessive schedule, which he did to go write down Takada's name in the real notebook. Gevanni followed him to his safe-deposit box and yanked the real notebook from it, and had THAT one copied so Mikami could write in it all he wanted and kill no one.
So how did Light lose? He picked a servant who had a fucktastic sense of timing.