The phrase "live by" means to adhere to or follow closely. When someone "lives by" their religious professions, it means they actively apply the principles, beliefs and values of their religion in their daily lives. 

"Live to"means to strive to attain a purpose or goal, as in "live to see another day" or "live to achieve one's dreams." "Live with" means coexistence or tolerance, such as "live with your mistakes" or "live with difficult circumstances."  "Live on" refers to survival or dependence, like "live on bread and water" or "live on someone's generosity."