Fitting a log (or generalized?) linear model I would like to fit a model of the form
$z = k x^\alpha y^\beta$
to some data I have (it's a spatial gravity model).  Now I know you could take logs of both sides and fit a linear regression
$\log z = \log k + \alpha \log x + \beta \log y$
But there are two problems:


*

*Does this violate assumptions of OLS regression if $\log x$ and $\log y$ don't have a normal distribution?

*Some of my $z$s are zero - how to deal with that?
EDIT
In response to whuber's question.  This is a model of how many people from each of a number of districts travel to an event.  $x$ is district population and $y$ is (mean) distance (from each district) - thus I expect $\alpha \approx 1$ and $\beta < 0$.  $z$ is sampled from a survey of under 1% of over 100k visitors.  I am interested in obtaining estimates for where the remaining 99% of visitors came from, for which I do not expect $z=0$ for any district, but in the limited sample data $z$ is sometimes zero because the survey people didn't happen to meet anybody from that district.
If anyone has comments about how the above modifies expectations about the error distribution on $z$ and hence the best way to fit the model I would be interested to hear it.  As I was expecting $a \approx 1$, I tried nonlinear least squares with $a$ fixed at 1, but that led to extremely bad variance on $k$ ($\sigma_k > k$).  I have since fitted GLM with a poisson link function which gives sensible $k$ and $\beta$ but $\alpha \approx 0.85$ - not great, but it's only a rough model so I can live with it.
 A: I would use a Poisson Quasi-MLE approach for a gravity model, especially with zeros in $z$. Code, intuition, and papers can be found at the Log of Gravity page. 
A: How to do this in any software is off-topic here, but there are statistical questions that transcend software choice. You need to work out what to do before you seek the code to do it. 
Question 1 is backwards. You can't fit your first form directly using standard regression functions or commands; you would need to turn to non-linear least squares. Both that and your second form imply assumptions about the error term, not the distributions of $x$, $y$, $\log x$ or $\log y$. As it happens, my wild guess with data like yours is that $\log y$ and $\log x$ will be closer to normal than the original variables; that does no harm, but it's not strictly relevant. The question is one for you, which is more realistic, the idea that errors are additive compared with $y$ or additive compared with $\log y$? 
Question 2 is enormously difficult. Zeros in the response $z$ imply to me turning to a generalized linear model, but there is a serious argument that if zeros arise for structural reasons you may need a more complicated model in which extra zeros are modelled as such. There is no agreed answer on applying a logarithmic transformation of a predictor when there are zeros present, but this is much discussed elsewhere on this site, notably in this FAQ
I don't know the current literature here, but these problems are basic and should be prominently discussed in the better papers. If not, shame on the practitioners! 
