www.exceptional.io is exactly what you're asking for. It costs $9 per month after a free trial.
Integration with PHP is done by including a wrapper and a couple of lines of code: https://github.com/ankane/exceptional-php/
It can also be integrated with javascript with a couple of lines:
<script type="text/javascript" src="http://js.exceptional.io/exceptional.js"></script>
<script type="text/javascript">
Exceptional.setKey('your_api_key');
</script>
Another option is New Relic, although it does a lot more than purely error logging - it provides all kinds of server monitoring metrics. https://docs.newrelic.com/docs/applications-menu/errors-dashboard