Kinda depends on the logic you are allowed to use, but in general:

NOT weather(snow, Thursday)

weather(sunny, Tuesday)

take_day_off(Tuesday) XOR take_day_off(Thursday)

take_day_off(x) IMPLIES (weather(rain, x) XOR (weather(snow,x))

try and find the missing information now, we know the weather on both days, s we need to know if the day was taken off on Thursday or Tuesday.

if Tuesday was taken off, this would mean that (weather(rain, x) XOR (weather(snow,x)) is true, which it is not since weather(sunny, Tuesday). So take_day_off(Tuesday) is false.

take_day_off(Tuesday) XOR take_day_off(Thursday) now says that take_day_off(Thursday) is true. a combination of NOT weather(snow, Thursday) and take_day_off(x) IMPLIES (weather(rain, x) XOR (weather(snow,x)) now says that the weather on Thursday was rain.

Sorry for the crappy notation, forgot how to make the logical symbols on here...