Last vs. The Last

• She was married last April. (NOT …in the last April.)
• I saw him last Tuesday. (NOT ….on the last Tuesday.)

The last

The last week refers to the period of seven days up to the moment of speaking or writing. Note that these time expressions are normally used with perfect tenses and with prepositions.

• We have lived here for the last three years. (= since three years ago)
• I have been busy for the last two months. (= for the two months up to now.)

Note that we generally say the last few days/weeks, not the last days/weeks.

• It has been raining for the last few days. (NOT — for the last days.)