Interpreting log-log regression with log(1+x) as independent variable Is interpreting log-log regression with log(1+x) as independent variable the same as having log(x) as independent variable? 1% increase in x results in beta% change in y?
What is x has both negative and positive values with the absolute value less than 1 so that 1+x are all positive?

I have always used the following intuitive explanation where log(1+x) (for positive x) is the transformation:
For small x (<< 1), $\log(1+x) \approx x$ . So the effects are still linear.
For large x (>> 1), $\log(1+x) \approx \log(x)$ . So the effects are multipliative.
Therefore, the transformation log(1+x) moves from being linear at one extreme to multiplicative at the other, and is somewhere between the two for "middle-sized" values of x.
By way of a concrete example, if the beta-value was +0.3, then this would be saying
"for small response variables, an increase of one unit of the predictor variable would lead to an additive increase of +0.3. For large response variables, an increase of one unit of the predictor would lead to a multiplication by 1.3 (or +30%). For response variables which are neither small nor large, the effect will be somewhere between the two."
