Linear Regression: units of intecepts given $x$ and $y$ as log function?

Question

Given a line $y = ax + b$, where $x = log(w)$ and $y = log(l)$. To me, the units of $b$ should be unitless as $y$ will be unitless. Is this right?

Answer 1

Yes. Formally speaking, you're not taking $y = \log (l)$ but $y = \log (l/l_0)$, where $l_0$ is one of whatever your unit of measure is - the argument of the log needs to be unitless. From the fact that you've chosen $l$ and $w$, probably "length" and "width", I'm guessing you mean for $l$ and $w$ to have units of length, so $l_0$ is one meter or foot or whatever. Similarly you really have $x = \log (w/w_0)$. But nobody ever spells this out.

The slope $a$ is also unitless, since $y$ and $x$ are both unitless, and $y$ and $ax$ must have the same units.