leave something out means omit it, not include it (e.g. In the sentence ‘She said that she was ill’, you can leave out the word ‘that’)
let somebody down means disappoint somebody because you didn’t do what they hoped (e.g. You can always rely on Tom. He’ll never let you down)
let somebody in means allow to enter (e.g. Can you let the cat in before you go to school?)
log in/ on means sign in/ out (e.g. I can’t log in to Facebook because I’ve forgotten my password.)
log out/ off means sign out (e.g. If you don’t log off somebody could get into your account.)
look after somebody/ something means take care of (e.g. I have to look after my sick grandmother.)