Why are elasticities defined as logarithmic derivatives? In my economics class, we often compute the elasticity of $Y$ with respect to $X$,
$$\eta = \frac{\partial \log Y}{\partial \log X}.$$
You can compute this from the slope of a line fit to a log-log plot. 

Why is it more natural to consider this quantity than the much simpler quantity
$$\eta' = \frac{\partial Y}{\partial X}$$
which is just as easy to measure? There seems to be some implicit assumption that $\eta$ is perhaps 'more applicable', maybe that it's less sensitive to the value of $X$ than $\eta'$ is, so it's more useful for extrapolation. For example, I've seen a demand curve extrapolated by assuming that $\eta$ is constant, yielding a power law. But why not assume $\eta'$ is constant, yielding a line? Neither seems particularly more natural to me.

Why is $\eta$ a more useful quantity than $\eta'$ in economics?
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 $$ El_X(Y)=\frac{d Y}{d X}\frac{X}{Y}=\frac{d\log(Y)}{d\log(X)}.$$