I have a connecting flight going from San Francisco to Beijing to Busan Korea.

As you stop over in Beijing for just three hours, you don't need to apply for the 72 hour free pass or an entry permit. When you pass through the immigration counter, the officer will stamp on your passport, which allows you to change the flights without the need to apply for an entry permit.

Are you going to stay in the airport transit area or wander about outside?

Thank you Michelle. That is exactly what did happen.