Where is vagrant's log file?

Vagrant does not keep any logs. The output of, for example vagrant up --debug, can be redirected to a file like

vagrant up --provision --debug &> debug_log

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Fragment from an IRC session in #vagrant at Freenode:

[18:29] <NikosA> ada: really, where is the debug "file" stored, by default? Isn't there any?

[18:29] <ada> vagrant does not write to log files

..

[18:29] <ada> virtualbox does

[18:29] <NikosA> I am even trying to >> debug_log in the command line, and it simply does not keep any of these valuable details.

[18:29] <NikosA> ok, so I'd check for the VBox logs?

[18:30] <ada> if you're using the vbox provider, yes

..

[18:31] <dtrainor> NikosA, redirect the output with &>