Are you sure you are not mis-reading something? That's not close at all. For example, if X= 53 and Y= 25, then (X-Y)/Y= 1.12 while ln(X)- ln(Y)= 0.751. Taking X and Y very large, say X= 100000 and Y= 800000 makes it a bit better: (X-Y)/Y= 0.25 and ln(X)- ln(Y)= 0.22. Is this supposed to be for X and Y very large? Or perhaps your book is requiring that X and Y be very close- that is that X-Y is small compared to Y?