Callback for job applicant
This looks pretty good, not much room for improvement I think. LinkedIn's JavaScript is calling linkedInApplySuccess, correct? All you have to do is specify data-success-callback, or something similar, in your HTML? 

The only minor thing I would do is instead of sending just the job ID, send the entire JSON object. As long as resources aren't an issue, it's always best to capture all data and disregard the stuff you don't use rather than only collect the stuff you're currently using; this is because requirements change, and in the future you may want to do something with the data you've been ignoring. For instance, you may want to calculate how many people are applying for jobs with the position "Chief Weasel Wrangler".

You can alter your code to something like this:

$.post("/php/cv_db/ajax_log_linkedin_application.php", {application: jsonReturnObj});


(application refers to "job application" not "computer application" – maybe too ambiguous)