Please look at the image below. We work with Jira Agile and sprints. We move issues through different columns in our work flow. But we ALWAYS forget to log how many hours we've worked on specific issues.
Is there any way to have Jira prompt me the "Log work" field when I move an issue between the fields "In progress" and "Ready for test"?

One way to achieve this would be to create a custom screen that contained only the log work field. Then associate that screen with the transition from 'In Progress' to 'Ready to Test'.
That way, when you transition an issue the custom screen will pop up. It would still be up to the individual to complete the field, but they would have no excuse for forgetting to do it.