A magna cum laude graduate of Auburn University, Logan joined Southern Progress in 1970. He was named president of their affiliate Oxmoor House in 1978. Southern Progress was acquired by Time, Inc. in 1985. Logan was named CEO of Southern Progress in 1986. In 1992, he became President and COO of Time Inc., and moved to chairman and CEO in 1994. In July 2002 he was named CEO of Time's newly acquired Internet service provider, AOL. In 2005, Logan took on a non-executive chairman role at Time Warner.
Logan has served as a director of the Birmingham Chamber of Commerce, as co-chair of the Civil Rights Institute in Birmingham, and as a board member of the Magazine Publishers of America and the National Book Foundation.
He and his sons Stan and Jeff have owned the Barons since September 2005. Their Birmingham Barons LLC is the 11th owner of the Barons.
In 2009 Logan was inducted into the American Advertising Federation's Advertising Hall of Fame. He was inducted into the Birmingham Business Hall of Fame in 2015 and the Alabama Business Hall of Fame in 2017.