Ultimately log files are useful for finding issues, especially early on in a project. And if they're not needed, they can be purged after. There is one caveat to this statement, too much logging can certainly be a problem in running full scale performance tests. In fact, if you find you can't get up to a decent load, it may be worth reviewing your level of logs and results listeners etc.
What I have added here is simply to download any logs you do have. To this end, I save the standard controller output from screen to a file and I specify a file for the jmeter log. These can then be copied up to the Jenkins workspace, for example.
JMeter logging levels can be controlled in the jmeter.properties file and this can sometimes be useful if you have unexpected issues showing up
To get the standard output to file I use tee. After some use with command line pipes, in various batches and Jenkins, I found a better solution was to build this straight into the controller script. So now, near the top of the script I use: