You can find the log within you Magento root directory under
var/log
there are two types of log files system.log and exception.log
you need to give the correct permission to var folder, then enable logging from your Magento admin by going to
System > Configuration> Developer > Log Settings > Enable = Yes
system.log is used for general debugging and catches almost all log entries from Magento, including warning, debug and errors messages from both native and custom modules.
exception.log is reserved for exceptions only, for example when you are using try-catch statement.
To output to either the default system.log or the exception.log see the following code examples:
Mage::log('My log entry');
Mage::log('My log message: '.$myVariable);
Mage::log($myArray);
Mage::log($myObject);
Mage::logException($e);
You can create your own log file for more debugging
Mage::log('My log entry', null, 'mylogfile.log');