If you install a game on steam on your PC will the game work if you log on to your steam on your mac

No, because you are not using Steam to play the game, you are only using Steam to verify the game and to use the multiplayer parts of it. If you have a Mac version DVD/CD of the same game then you could still use it on steam if you register it on Steam.
I THINK if you bought the Windows game on steam they will let you download the Mac version on the same machine. They have the full instructions on their web site.