Where does homebrew log?
~/Library/Logs/Homebrew/

Homebrew creates log files in your Library folder at: ~/Library/Logs/Homebrew/

You can view the log files by holding down Option and using the Finder menu item: Go > Library, then navigating to Logs > Homebrew.

Alternatively, you can use the Console.app application to browse to the log files.

Homebrew History

The default creation of individual log files was added during 2013 to Homebrew.

Homebrew issue #10430 talks about logging and build errors. The issue report mentions no log file is kept but explains that a log can be created with the command format:

brew install &lt;formula&gt; 2&gt;&amp;1 | tee install.log