power regression when the power is a variable I have this function :
$y = x^\alpha$
using log:
$\ln(y) = \alpha\,\ln(x)$
Now, $\alpha$ itself can be decomposed and considered as a function of two variables $w_1, w_2$. We have:
$\alpha = \beta_0 + \beta_1 w_1 + \beta_2 w_2$
Altogether:
$\ln(y) = (\beta_0 + \beta_1 w_1 + \beta_2 w_2)\,\ln(x)$
My question is how to consider this alpha, and how to include it in the $y$ regression.
Let's say I have 30 values for $w_1, w_2$ and 30 values for $x$.
My thinking (which is probably wrong):
I would like to do a regression with $w_1$ and $w_2$. But a regression of what on what?
Then incorporate the results of this regression, in the second one, the $\ln(y)$.
I am a bit lost on how to approach this problem. Thank you for your help !
EDIT:
$x, w_1, w_2, y$ are all different variables (columns). Each of these variables has 30 observations (rows). $w_1$ and $w_2$ do not relate directly to $y$, but relate to $x$ and hence relate to $y$. 
In other words: $w_1$ and $w_2$ are measuring education of a population. 
Alpha represents the productivity of this population in the field of research.
$x$ represents the number of researchers.
$y$ represents the number of patents.
 A: *

*If you assume that the error in the observed $y$ arises only through noise in $\alpha$ (rather than say additive error), then you might indeed use logs to get to
$\qquad \ln(y) = \alpha\ln(x)$, i.e.
$\qquad \ln(y)=(\beta_0 + \beta_1 w_1 + \beta_2 w_2+\epsilon)\ln(x)$
In that case, you could just divide through by $\ln(x)$, and then letting $y_1 = \ln(y)/\ln(x)$, we get:
$\qquad y_1=\beta_0 + \beta_1 w_1 + \beta_2 w_2+\epsilon$
which reduces to a simple linear regression.

*On the other hand if you assume a multiplicative error:
$\qquad y = x^\alpha\cdot\exp(\eta)$
then taking logs would give:
$\qquad \ln(y)=(\beta_0 + \beta_1 w_1 + \beta_2 w_2)\ln(x)+\eta$
and if you assume that $\eta$ has constant variance and mean $0$ (define $v_0=\ln(x)$, $v_1=w_1\ln(x)$, $v_2=w_2\ln(x)$, and $y_2=\ln(y)$), then you might proceed by fitting an ordinary regression (with no intercept):
$\qquad y_2=\beta_0 v_0 + \beta_1 v_1 + \beta_2 v_2 + \eta$

*Then again, if you do assume that there's an additive noise (of approximately constant variance) 
$\qquad y = x^{\beta_0 + \beta_1 w_1 + \beta_2 w_2}+\xi$
then you might instead consider using nonlinear least squares to fit the model.
There are of course any number of other possible views of how the noise in the response would come into the relationship.

The edit changes the question substantially.


*

*Clearly the relationship between $y$ and $x$ cannot be noiseless, so the model formulation is simply wrong.

*$x$ and $y$ are counts; you should probably not simply take logs but consider models suitable for counts (with log-link presumably) - perhaps Poisson regression or negative binomial regression for example); if you do taje logs and use regression anyway, you should not expect constant variance on the log scale. 
