Google analytics events not showing up

In google-analytics, for the correct website, look on the "realtime/events" page and see if events are appearing there (they should appear here within a few seconds, after being triggered on your website).

If not, you might have added a filter, to filter out events generated by traffic from your own IP. You can find this filter under "admin" and then "all filters". If so, remove it (or temporarily change the IP-address, so it doesn't match). After that, generate a few events on your website and look again on the google-analytics page, under realtime traffic / events page.

If they work under realtime, it should also work in non-realtime (but it might take 24-48 hours to appear there!)

Not sure if this just a typo in your example, but you are missing the event handler in your link:

<a href="http://example.com/" target="_blank" class="app-logo" onclick="ga('send', 'event', 'Link', 'Click', 'example.com');"></a>

The event handler ("onclick") is the bit that specifies which action shall trigger the javascript code. If you just dump the event call in the HTML it will not even be recognized as JS.