Your query isn't using the index because you're making MySQL convert the datetime field to a varchar, and then that against the "like" operator.
What you want is probably something like "where date_live>='2004-09-22' and date_live<'2004-09-23'", and then let your app deal with displaying only the date. (In this approach, the constants will be converted from char to datetime before the comparisons, as opposed to the "like" example where for all MySQL knows you're examining the fourth character, so it has to convert all of the dates to chars)
At least, that's my guess. I'm an MS SQL guy
Cheers
-b