If you have Try and cache block and loggin error manually than you can try this :
$mylogfilepath = plugin_dir_path(__FILE__).'debug.log';
$errormessage="SOME ERROR".PHP_EOL;
error_log($errormessage, 3, $mylogfilepath);
Place debug.log file directly inside your plugin. and it will log error to that file. third attribute in error_log set path of file in which log should be recored.