If you're handling sensitive data and you don't want exceptions logging details such as variable contents when you throw them, you may find yourself frustratedly looking for the bits and pieces that make up a normal stack trace output, so you can retain its legibility but just alter a few things. In that case, this may help you:

<?php

function exceptionHandler($exception) {

    // these are our templates
    $traceline = "#%s %s(%s): %s(%s)";
    $msg = "PHP Fatal error:  Uncaught exception '%s' with message '%s' in %s:%s Stack trace: %s   thrown in %s on line %s";

    // alter your trace as you please, here
    $trace = $exception->getTrace();
    foreach ($trace as $key => $stackPoint) {
        // I'm converting arguments to their type
        // (prevents passwords from ever getting logged as anything other than 'string')
        $trace[$key]['args'] = array_map('gettype', $trace[$key]['args']);
    }

    // build your tracelines
    $result = array();
    foreach ($trace as $key => $stackPoint) {
        $result[] = sprintf(
            $traceline,
            $key,
            $stackPoint['file'],
            $stackPoint['line'],
            $stackPoint['function'],
            implode(', ', $stackPoint['args'])
        );
    }
    // trace always ends with {main}
    $result[] = '#' . ++$key . ' {main}';

    // write tracelines into main template
    $msg = sprintf(
        $msg,
        get_class($exception),
        $exception->getMessage(),
        $exception->getFile(),
        $exception->getLine(),
        implode(" ", $result),
        $exception->getFile(),
        $exception->getLine()
    );

    // log or echo as you please
    error_log($msg);
}

?>

If you're not a fan of sprintf() or the duplicate $exception->getFile() and $exception->getLine() calls you can of course replace that as you like - consider this a mere compilation of the parts.