This font is very similar to Clarendon, News 701, and Century Schoolbook, but I can't find an exact match. If anyone can help me with this, I'd appreciate it. Thanks.

Clarendon  Suggested by Sneggleblech 
New Century Schoolbook Bold  Suggested by jerseygirl 
Century Schoolbook  Suggested by Parker76 

New Century Schoolbook wasn't created until 2007, and this image comes from 1984. I already said the original version of Century Schoolbook wasn't a match anyway.

I haven't bumped this up in quite some time; does anyone know of a match for this logo?

Font Meme says it's Clarendon

10,000% percent, the right answer is Century Schoolbook.

Keep in mind that Karate Kid came out in 1984, way before digitized fonts existed. It's likely that the designer who did the Karate Kid logo used a version of Century Schoolbook that never got digitized. Keep in mind that back then, there were tons of different type foundries, each with their own version of Century Schoolbook that they sold, and each having slight differences one from the next.

So ya, it's possible that the Karate Kid logo doesn't match any currently existing digitized versions of Century Schoolbook ... but nonetheless, the correct answer is still 'Century Schoolbook'.