Does CBD stand for central business district? Los Angeles is a huge, spread-out area, and for many the name is used to encompass not just the City of Los Angeles, but the wider County of Los Angeles, and even nearby counties (e.g., Orange County, where Disneyland is located). There is not one central business area. There are many. L.A. County has something like 75 different cities. I grew up in suburban Van Nuys and almost never went to downtown L.A. Downtown is more lively now, and we found it interesting to stay there (at the historic Millennium Biltmore, where the Academy Awards ceremony took place in the 1930s), but most tourists are looking more for attractions near the ocean, at amusement parks or in Hollywood (using Hollywood as a general term; the best studio tours are probably Warner Brothers in Burbank and Sony (formerly MGM) in Culver City).

You need to determine what sites you are going to see, and how you'll get around (public transportation, taxi or rental car). Hotel within the airport? Uh, no. But there are lots of hotels near LAX. You can get good prices through Hotwire or Priceline. Parking can be expensive and it's not a particularly convenient place to use as a base to see Disneyland, Hollywood, etc. (especially using public transportation),though it can be done. Santa Monica, along he ocean north of LAX, is a pleasant, though not inexpensive, area.

I'd start by looking at a guide book (maybe find one online), and a map. Just think of Los Angeles as more of a state than a city with one central area you'd want to be based in.

Keep in mind that L.A. traffic can be really bad throughout the day (hardly my idea of relaxing) and you may have to pay a tidy sum to park at many hotels. On the plus side, for your price range (assuming it's in U.S. Dollars), you should be able to find lots of places. Mybe even the 5-star Hotel Bel Air or Beverly Hills Hotel or Shutters on the Beach in Santa Monica (but don't quote me on that). You could get some additional ideas from sites such as Tripadvisor.com. I assume you've already determined how much extra, if any, you'll need to pay in order to break your trip in L.A.