Is there a way for a developer to log into a dev task so time for that specific task is tracked?

[Aaron T]

Right now a developer will give us 50 hours estimation for project which at $10/hr we can set a budget for $500.

The problem

They start the project and we have no way to track how much actual time is being spent on a daily/weekly basis.

And the developers dont have a way to log into a project and then log out to start and stop the project timer.

If we can find a solutions that they can use to be more accountable I know that we will increase productivity and use far less resources