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I hope the below command will help you

ls | grep -e "`date "+%Y%m%d"`" | grep ".log"

If the date (in format YYYYMMDD) is not part of the log file then,

find . -mtime -1 -name "*.log"

Unix allows you to give three levels of permissions to the file.
a. Owner
b. Group
c. Others

When we do a ls -lt, the following is a sample output.

-rwxrwxrwx 1 vijay ...

The .profile will be available in '/' . Check it there.