How to interpret a log transformed (x+c)? [duplicate]

I need to use log-log regression and because I have lots of zero values I tried to add a very small constant c=8E-12 to x and it works pretty good. Xs are very small probabilities.

lnY= a + b ln (x+c)

But how do I interpret that model? Thanks

• Why don't you tell us something about the original model and data, why "you need" to use log-log regression, and how the results will be used? – Mark L. Stone Apr 25 '16 at 21:06
• I´m trying to create model for my bachelor thesis. I use gravity model to explain influence of religion on international trade and log-log regression is the very simple solution. – David Apr 25 '16 at 21:13
• – whuber Apr 25 '16 at 21:13
• Oh, thanks. I checked all of them, but now I see I missed few. – David Apr 25 '16 at 21:19
• "Generally, using log(1+y)log(1+y) and then interpreting the estimates as if the variable were log(y)log(y) is acceptable when the data contain relatively few zeros" says Wooldridge. But what if I have lots of zero values? – David Apr 25 '16 at 21:21

The problem is, your $c$ is a guess and may bias the results even if your assertion is true. You may want to try several values.
Your model is related to $Y = e^a (x+c)^b \times \epsilon$.
If you intend to fit $Y = e^a (x+c)^b + \epsilon$ then how you fit the model matters. You maybe could estimate $c$ as a parameter but this could be problematic and may not be worth the bother.