mkdir /tmp/test
gedit /tmp/test/test.sh
#! /bin/bash

0-59 * * * * * /tmp/test/test.log

mkdir /tmp/test
gedit /tmp/test/test.sh
#! /bin/bash
DAY=25;
0-59 * * * * * /tmp/test/$25.log

DAY25; 
crontab: error on previous line; unexpected character found in line.
crontab: error detected in input, no crontab file generated.

mkdir /tmp/test
gedit /tmp/test/test.sh
#! /bin/bash
DAY=`date +%d`;
0-59 * * * * * /tmp/test/$DAY_test.log

DAY=`date +%d`; 
crontab: error on previous line; unexpected character found in line.
crontab: error detected in input, no crontab file generated.

mkdir /tmp/test
gedit /tmp/test/test.sh
#! /bin/bash
0-59 * * * * * /tmp/test/$(date +%d)_test.log

sh: syntax error at line 1: `(' unmatched

touch /tmp/test/`date '+%T'`.log

Содержимое test.sh:

#! /bin/bash
0-59 * * * * touch /tmp/test/`date '+%T'`.log

crontab /tmp/test/test.sh
crontab -l
#! /bin/bash
0-59 * * * * touch /tmp/test/`date '+%T'`.log
You have mail in /var/mail/oracle

Содержимое /var/mail/oracle:

touch /tmp/test/$(date +
sh: syntax error at line 1: `(' unmatched