Solution 1:

It depends on the exact model.

A number of Logitech mice/keyboards use a "universal" dongle. For these USB dongles there is a utility that you can download from the Logitech web-site with which you can re-pair dongles and devices. This works for any mouse/keyboard that uses the universal dongle.

For non-universal dongles some can be re-paired with another keyboard/mouse of the same model.
For information on how to do that (and if it is possible at all for your specific model) you will have to check the Logitech support pages on their web-site.