grep between date ranges in a log
A Perl solution:
perl -wne 'print if m/(?<!\d)(20\d{8})(?!\d)/ && $1 >= 201211150821 && $1 <= 201211150824'(It finds the first ten-digit integer that starts with 20, and prints the line if that integer is within your range of interest. If it doesn't find any such integer, it skips the line. You can tweak the regex to be more restrictive about valid months and hours and so on.)
You are looking for the somewhat obscure 'csplit' (context split) command:
csplit '%201211150821%' '/201211150824/' file
will split out all the lines between the first and second regexps from file. It is likely to be the fastest and shortest if your files are sorted on the dates (you said you were grepping logs).