I had this issue and was able to resolve it. For us, the issue was that the PostgreSQL database was not installed. You can verify this by searching the server for PostgreSQL. If it is not found on the server, you can download it using a link that is in this page I link below. After installing it on the server, you then have to configure it with the instructions shown in this page.
You're not done yet. Now you have to set up the logging service at the Command Prompt on the server as shown in this article. This part is a bit more tricky because you need to know the accounts and passwords, and be familiar with navigating at the command prompt. However, after a couple failed attempts, I was able to execute the command successfully, and THEN I was able to start the Qlik Logging Service. Good luck!!