Are these two queries logically equivalent?
Whether the two queries you posted are logically equivalent is irrelevant; you shouldn't use either of them. I will try to steer you away from a couple of things:


Whenever possible, try to avoid applying functions to columns. It's always as good, and mostly better, to keep those calculations against constants and not columns - this can destroy SARGability and render indexes on those columns useless. In this case, I much prefer query 2, especially if LogDateTime is indexed (or might ever be).
I don't like the shorthand date math and I recommend against it. Sure, it's faster to type, but try that with a DATE data type and you will get an ugly error. Much better to spell it out, e.g.:

WHERE LogInsertTime &lt; DATEADD(DAY, -7, @DateTime);