I assume that you use Slackware 13.37. This version is to old, so that the pre-installed free drivers don't work with your videocard (I had the same problem with my HD6870). I would recommend to install the proprietary AMD drivers.
To do that log in as root and launch the following commands:
wget -c http://www2.ati.com/drivers/linux/amd-driver-installer-12-6-x86.x86_64.run
This will download the driver. To install it use
and follow the onscreen instructions. Then you have to create an initial configuration:
If you now reboot
and use startx again it should work.