You can, if the logs are conducted. Still need a backup at some date + binary logs from this date... (from "for performance reasons" - they usually turn off).
Your actions:
1)need a backup of the database for some date: date1
2)the base is removed in date: date2
3)mysqlbinlog -s -d db_name -u user_name --start-datetime="date1" --end-datetime="date2" -t mysql-bin.[0-9]* > out.sql
4)restore the backup database for date1
5)add the inquiries from a binary log in the interval date1 date2...:
mysql db_name < out.sql
Or try to restore using Recovery Toolbox for SQL Server. This is a very strong utility and repairs efficiently and quickly. Can use.