This is a little different from Banthar's solution, but it will work with versions of find that don't support -newermt and it shows how to use the xargs command, which is a very useful tool.
You can use the find command to locate files "of a certain age". This will find all files modified between 5 and 10 days ago:
find /directory -type f -mtime -10 -mtime +5
To then search those files for a string:
find /directory -type f -mtime -10 -mtime +5 -print0 |
xargs -0 grep -l expression
You can also use the -exec switch, but I find xargs more readable (and it will often perform better, too, but possibly not in this case).
(Note that the -0 flag is there to let this command operate on files with embedded spaces, such as this is my filename.)
Update for question in comments
When you provide multiple expressions to find, they are ANDed together. E.g., if you ask for:
find . -name foo -size +10k
...find will only return files that are both (a) named foo and (b) larger than 10 kbytes. Similarly, if you specify:
find . -mtime -10 -mtime +5
...find will only return files that are (a) newer than 10 days ago and (b) older than 5 days ago.
For example, on my system it is currently:
$ date
Fri Aug 19 12:55:21 EDT 2016
I have the following files:
$ ls -l
total 0
-rw-rw-r--. 1 lars lars 0 Aug 15 00:00 file1
-rw-rw-r--. 1 lars lars 0 Aug 10 00:00 file2
-rw-rw-r--. 1 lars lars 0 Aug 5 00:00 file3
If I ask for "files modified more than 5 days ago (-mtime +5) I get:
$ find . -mtime +5
./file3
./file2
But if I ask for "files modified more than 5 days ago but less than 10 days ago" (-mtime +5 -mtime -10), I get:
$ find . -mtime +5 -mtime -10
./file2
