You Do Need power plug adapter while traveling from Luxembourg to Falkland Islands.

Here is Why You Need An Adapter

The outlet and plug type used in Falkland Islands is G and in Luxembourg are C and F. This is how they look like.

Luxembourg

In Luxembourg, outlets of type C and F are in use.

Type C

Type F

Falkland Islands

In Falkland Islands, outlets of type G are in use.

Type G