Mello explanation:
Mello kidnapped Takada and put her in a situation where it may be revealed that she was the one writing the names in the death note. This caused Mikami, who couldn't contact Light or know that Light was able to kill Takada, to go and get the real notebook he had out of the safe deposit box when he wasn't suppose to. This is how Gevanni learned that the notebook Mikami was carrying was fake.