The most common way to keep records is a logbook. A logbook can be as simple as a notepad, where all data are entered manually in writing and as complex as dedicated software packages that are available commercially. It is possibly a good idea to have an electronic logbook and to print out hard-copies when needed. If you decide to have an electronic logbook, please make sure you make back-ups at regular intervals.
Most contractors will offer to keep the logbook for their customers. Please keep in mind that you as the operator (or owner) of the equipment will still be responsible for compliance, so if the logbook is found inaccurate, you will be the one to get the fine. It may also complicate matters if you use more than one contractor, or if you change contractors.
In some countries, you may need approval by the authorities for the type of logbook you want to use.
To download an example of logbook: Tools section