Let X = 10^y.
So the question becomes Is y close to -2 or to -3?
(1) y is closer to -4 than to -1.
(2) Y is closer to -3 than to -1.

Consider statement (1) alone.
If y is closer to - 4 than to -1, then there can be two cases.
Either y < -4 or - 4<y<-1.
If y < -4 obviously it will be closer to -3 than to -2.
If -4<y<-1 and y is closer to - 4, it means distance between y and -4 on the number line is less than the distance between y and -1.
Now distance between -4 and -1 is -1-(-4) = 3.
Half of this is 1.5.
So y-(-4) < 1.5
Or y+4<1.5
Or y < -2.5.
Take an example. Let y = -2.6 which is less than -2.5.
Distance between -2.6 and -2 is -2- (-2.6) = 0.6
Distance between -2.6 and -3 is -2.6 - (-3) = 0.4
So we see that -2.6 is closer to -3 than to -2.
So we can conclude that any number closer to -4 than to -1 is also closer to -3 than to -2.
So (1) is sufficient.
Consider (2)
Again there can be two cases.
First that y < -3 and second -3<y<-1.
In the first case it will be closer to -3 than to -2.
In the second case we will have that distance between y and -3 is less than the distance between y and -1.
Distance between -3 and -1 = 2
Half of this is 1.
So y-(-3) < 1
Or y<-2
Take y = -2.2 for example.
Dist between -2.2 and -2 is 0.2. and between -2.2 and -3 is 0.8.
Here y is closer to -2 than to -3.
Since nothing definite can be said we say that (2) is not sufficient.

The correct answer is A.