Location of Drush Error log
The Drush logs are emitted only to the console.  Use --debug to get more information about what is happening.

Your ISP is probably deciding that your Drush command is taking too long, and are killing the process.  Ask support at your ISP to determine if this is the case, and whether there is a way to increase the limit.   Drush does not have any support this sort of limitation; it assumes that it is running in a shell with no time limits. 

PHP can also be configured to apply a maximum timeout to your requests. To find out what php.ini Drush is using, run drush status. One of the lines will provide the full path to your php.ini file.  Ideally, you should use a different php.ini for Drush than you do for your web server.  Set the environment variable PHP_INI to point to the php.ini file that Drush should use. Copy your webserver php.ini to serve as a starting point, and then customize to suit.

The variable you are looking for is max_execution_time.