How to interpret a log transformed (X+1)? Assume that I have a log transformed model as follows:
Model 1: $Y = a + b\ln(X)$.  Interpretation: a 1% increase in $X$ is associated with an average $b/100$ units increase in $Y$.
If I add $1$ to $X$ to avoid having $0$ values and get:
Model 2: $Y = c + d\ln(X+1)$
Should I interpret the model as "a 1% increase in $(X+1)$ is associated with an average $d/100$ units increase in $Y$?" Or there are some better ways to interpret the model? Thanks.
 A: You could, but it's not a very intuitive thing.
Consider for simplicity, starting at $c=0$ and $d=1$.
If $x$ is 0.01, then $y = \ln(1+x) \approx 0.01$ and a 1% increase in $(1+x)$ yields about a doubling of $\ln(1+x)$ (and hence, $y$) to  $y\approx 0.02$ which is about a 100% increase in $y$. Meanwhile if $x$ is 2 it's a little less than 1% increase in $\ln(1+x)$ (and hence $y$), and if $x=100$, it's close to a 0.2% increase in $\ln(1+x)$.
Changing $d$ from $1$ to something else (while $c$ remains at $0$) doesn't change this effect on $y$, because the relative increase in $y$, i.e. (ynew-yold)/yold is unaffected by the value of $d$.  However, changing $c$ does affect it, because the old value is on the denominator, and once $c$ is non-zero, then $d$ also matters.
A: The interpretation of the model should depend partly on the range of values of  $x$ and on how it is to be applied.  If most of the $x$ values are large, say more than 100, and if it is to be used to predict $y$ corresponding to such large values of $x$, then a good approximate interpretation would be: a 1% increase in $x$ is associated with an average $d/100$ units increase in $y$.  For $x > 100$ the proportionate difference between $x+1$ and $x$ is small.
If however the model is to be used to predict $y$ corresponding to small values of $x$ then this approximation would be unhelpful and your interpretation would be more appropriate, although as Glen_b says it's not very intuitive.
If most of the $x$ values are small, then a better way to avoid the zeroes might be to add a different constant, much less than 1.  
