I presume with log off you mean hibernate or similar. MIne turns the monitor off after 30 minutes.
This is for XP, rightclick anywhere on your desktop and select properties and then the screensaver tab. On the bottom you'll find the power settings. Press the power button and you can configure your computer to do anything you like except make you a cuppa tea. Note some of the custom settings may suit you better. If you want to be in complete control set everything to 'never'. Have fun.