Why did the navy devote an aircraft carrier to the search for AE?
Briefly:
When Earhart went missing the Chief of Naval Operations told the Commandant of the 14th Naval District at Pearl Harbor, Rear Admiral Orin Murfin, to do whatever was "practicable." Everybody knew that what was needed was aircraft but Murfin didn't have any aircraft carriers or battleships or cruisers. Pearl was not yet the home of the Pacific Fleet. The only aircraft Murfin had were PBYs so he sent one south to help with the search, but it had to turn back due to weather.
Colorado, with three float planes, was in Honolulu on its annual NROTC training cruise. Murfin got permission to commandeer the battleship for the Earhart search. He told the CNO that if he wanted more airplanes than that he'd have to send a carrier. The CNO was not at all sure he wanted to send a carrier but, just to be safe, he sent a message telling
Lexington to stand by for a possible mission. Somebody leaked the alert to the press and the the next day the papers were full of how the Navy was about to send Lexington to find Amelia. At that point the CNO was stuck. He had to send Lexington or look like he was callously abandoning Amelia Earhart to her fate.
Like much of the Earhart search,
Lexington's involvement was a comedy of errors.
It's all in
Finding Amelia.