How to migrate database without logs
You may want to move log tables from main database to specially created database via settings in civicrm.settings.php.  Of course, new empty database has to be created first. After that please find this in civicrm.settings.php: 

/**
 * CiviCRM Logging Database
 *
 * Used to point to a different database to use for logging (if desired). If unset defaults to equal CIVICRM_DSN.
 * The CIVICRM_DSN user needs to have the rights to modify the below database schema and be able to write to it.
 */
if (!defined('CIVICRM_LOGGING_DSN')) {
  define('CIVICRM_LOGGING_DSN', CIVICRM_DSN);
} 


when replace 

  define('CIVICRM_LOGGING_DSN', CIVICRM_DSN);


to 

  define('CIVICRM_LOGGING_DSN','mysql://DBUSER:DBPASSWORD@DBHOST/LOGS_DBNAME?new_link=true');


not sure if "new_link=true" is needed though. This automagically moves all LOG_ tables from the main database to  new database. Thus all LOG_ tables in main database could be deleted; main database is clean now and could be moved to new server (guess it's worth making same trick at new server). In theory, there could be some performance benefits as well, but not sure.