Equation for the equilibrium constant for the hydration of carbonyls using UV spectroscopy
The law is not as you state although it often gets reported as this. The '$A$' you quote is the optical density. The Beer-Lambert law is $I_{tr}=I_0e^{-\epsilon_\lambda [C]\ell}$ where $I_{tr}$ is the intensity of transmitted light for a molecule at concentration $[C]$ at wavelength $\lambda$ and cell path length $\ell$, and $\epsilon_\lambda$ is the extinction coefficient at wavelength $\lambda$. 

From the definition you can see where the expression you ask about comes from even with their different notation; it looks as if $w=-{\epsilon_\lambda [C]\ell}$ where ${\epsilon_\lambda [C]\ell}$ is the optical density.