Those head badges were used through 1983. Not sure of the first year for them, but they go back to around 1978, near as I can tell. Mid-1970s used the "north star" headbadge. Personally, my guess is 1980. The best way to determine the age would be to check the two letter date code on the back on the rear derailleur and compare it to the charts here;
http://www.vintage-trek.com/component_dates.htm . You can increase your confidence level by checking out the date codes on other components, but not the front derailleur, which is a obvious replacement!
Please report the results, so that I can add your bicycle to my Centurion Serial Number database. It's the first Custom that has shown up!