Why are elasticities defined as logarithmic derivatives?
If I understand your question, first the elasticity haven't units. The problem with $\partial Y/\partial X$ is that if you change the measure units the result is different. Is less problematic to express it in percentages:
$$ El_X(Y)=\frac{\Delta Y/Y}{\Delta X/X}$$
if $\Delta Y \to 0$ when $\Delta X \to 0$ then you obtain $$El_X(Y)=\frac{dY}{dX}\frac{X}{Y}$$ if $Y=Y(X,W,...,Z)$ you change $d$ for $\partial$.

Second, like you know $d \log(x)=\frac{dx}{x}$ thus $$ El_X(Y)=\frac{d Y}{d X}\frac{X}{Y}=\frac{d\log(Y)}{d\log(X)}.$$