Are you saying that someone reputable started with this step, with no explanation? That's not very good communication!
It does look like
@AvgStudent's idea is probably more or less the thinking that lies behind this, but it needs to be stated more clearly.
The initial inequality will be true when the numerator and denominator have the same sign, or when the numerator is zero.
Since
log2x−2 is an increasing function that is zero when
x=4, the sign of
log2x−2 is the same as the sign of
x−4.
Since
log2(x−2) is an increasing function that is zero when
x=3, the sign of
log2(x−2) is the same as the sign of
x−3.
We also have the condition that
x>2 so that both logs exist, which they have failed to mention.
So it is true that (apart from
x>2), the second line is equivalent to the first; but I see no reason to actually write that second line, when the next step (as I am guessing it should be) will be to use the facts I stated, rather than the second line which is based on those facts.
But it would have been far better if you had shown us the entire solution, not just these two lines, because then we could see where they are heading and hopefully more quickly understand why they are saying this. Please don't do this to us in the future. Show the entire problem and the entire work, even when you are only asking (you think) about one piece. Context often makes a big difference.