Dtyjlui

On a default installation the cron jobs get logged to

/var/log/syslog

You can see just cron jobs in that logfile by running

 grep CRON /var/log/syslog

If you haven't reconfigured anything,the entries will be in there.

You can create a cron.log file to contain just the CRON entries that show up in syslog. Note that CRON jobs will still show up in syslog if you follow the following directions.

Open the file

/etc/rsyslog.d/50-default.conf

Find the line that starts with:

#cron.*

uncomment that line, save the file, and restart rsyslog:

sudo service rsyslog restart

You should now see a cron log file here:

/var/log/cron.log

Cron activity will now be logged to this file (in addition to syslog).

Note that in cron.log you will see entries for when cron ran scripts in /etc/cron.hourly, cron.daily, etc. - e.g. something like:

Apr 12 14:17:01 cd CRON[14368]: (root) CMD (   cd / && run-parts --report /etc/cron.hourly)

However, you will not see more information about what scripts were actually ran inside /etc/cron.daily or /etc/cron.hourly, unless those scripts direct output to the cron.log (or perhaps to some other log file).

If you want to verify if a crontab is running and not have to search for it in cron.log or syslog, create a crontab that redirects output to a log file of your choice - something like:

01 14 * * * /home/joe/myscript >> /home/log/myscript.log 2>&1

This will redirect all standard output and errors that may be produced by the script that is run to the log file specified.

Sometimes it can be useful to continuously monitor it, in that case:

tail -f /var/log/syslog | grep CRON

You can also direct the output of the individual cronjobs to their own logs for better readability, you will just need to append the output of date somewhere.

 0 15 * * *    /home/andrew/daily-backup.sh >> /var/log/daily-backup.log 2>&1

This is a very old question, but none of these answers seem satisfactory.

First make your cron job run every minute, then run cron as non-daemon (temporarily, just kill any crond that may have already started) with test logging:

crond -nx test

And see the log of your program execution flowing through your terminal.

It is in /var/log/syslog by default.

But it can be set up to create a separate cron.log, which is more useful.

This Q&A describes the process:

16.04: How do I make cron create cron.log and monitor it in real time?

Also in this answer is the instructions to create a wcron command that displays it is near-real-time. Plus, it links to another answer,

How to change cron log level?

that shows how to change the log level to include more than just the start of jobs - level 15 will show errors and end time, also.

You could redirect the output of cron to a tmp file

Such as:
00 11 07 * * /bin/bash /home/ubuntu/command.sh > /tmp/output 2>&1

Error and normal output, both will be redirected to the same file

If you have systemd installed on your system, you could display cron job log by using the journalctl command.

For example, on my Ubuntu 17.10:

journalctl -u cron.service