。
I shouldn't have logged out.
Literal
Log-out-do [explanatory-negative-past-んじゃなかった] [emphasis-よ].
The pattern ~んじゃなかった expresses regret over a past action: 'I shouldn't have X-ed,' or more literally 'it was not the case that I should X.' It's the negative past form of the explanatory ~んだ (のだ). Another way to say this is ~なければよかった. ログアウトする is a する verb built on the loanword ログアウト.