Two ways to log jobs - both equally effective

You can either pipe the output if you are running batch (.bat) files or

In the jobs steps - advanced use the output file option.
If you have multiple job steps
 First step set the overwrite option
 Subsequent steps set the append option

Database dumps / log dumps
Make the database dump overwrite and then for log or differential dumps append so that you have a full job history in a single file

Don't forget to include the logs directory in the server backup schedule!

One word of warning...Permissions - the account that SQLServerAgent is running under will need permission to create files in the log directory.
All jobs for a server should be placed in the same log directory - and don't forget to occasionally check the contents of the log files - since that's how you will find out about problems!