I have an equation with logs in it. Specifically the equation takes the form:
$log(Y)=\alpha+\beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3$
Where $X_1$ through $X_3$ are variables. The trouble is that Y and each of the X variables have dimensions. Is there a way to rewrite this equation in terms of dimensionless quantities? I am running into the issue at the moment that since my variables have dimensions I find that they system behaviour depends on the type of dimension chosen!
I have tried writing each variable in terms of a dimensionless factor multiplied by a factor with units. Doing this we have that:
$log(\tilde{Y} \times \chi ) = \alpha +\beta_1 \tilde{X_1} \psi + \beta_2 \tilde{X_2} \eta + \beta_3 \tilde{X_3} \nu$
Where $\chi$, $\psi$, $\eta$ and $\nu$ are dimensionless, and the terms with a $\tilde{}$ are with units.
From here I am uncertain how to proceed in order to work with something without units. Is it possible to using this method and equation type? If not, please could you advise me on how to proceed?
Thanks,
Ben