Often in textbooks and online I see thte definition of an elasticity to be as follows:
1) $\epsilon = \frac{\frac{\Delta Y}{Y}}{\frac{\Delta X}{X}}= \frac{\partial \log Y}{\partial \log X}$
Thus, when using an elasticity along with a percentage change in X to estimate a percentage change in Y, it would seem to make sense that one would multiply the percentage change in X by the elasticity as below:
2) ${\frac{\Delta Y}{Y}}=\epsilon\cdot(\frac{\Delta X}{X})$
However, in one instance I've seen mention that it is more accurate to estimate the percentage change in Y as follows:
3) ${\frac{\Delta Y}{Y}}=(\frac{X_2}{X_1})^\epsilon-1$
I'm seeking to apply elasticities derived by estimating log-log regression models and would appreciate some guidance as to whether formula #2 or #3 is more appropriate when evaluating impacts on Y resulting from changes in X.
Can someone please explain why formula #3 may be more appropriate than formula #2? And how does one get mathematically from formula #1 to formula #3?