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?

$$\begin{array}{}
\frac{y^\prime/y}{x^\prime/x} &=& a& \quad \text{if $y=x^a$}\\
\frac{y^\prime/y}{x^\prime/x} &=& a \frac{x}{1+x}& \quad \text{if $y=(1+x)^a$}\\
\end{array}$$

You can interpret it as following: a $q\%$ change in $x$ leads to a $q \frac{x}{1+x}\%$ change in $(1+x)$.
If $x$ is large they have a similar interpretation but not when $x$ is small.
For instance, when $x=1$, then a doubling of $x$
is the same as a $50\%$ increase of $1+x$. The $x$ changes from $1$ to $2$ and the $x+1$ from $2$ to $3$.
When $x=100$, then a doubling of $x$
is nearly the same as a doubling of $1+x$. The $x$ changes from $100$ to $200$ and the $x+1$ from $101$ to $201$.

Or the other way: a $q\%$ change in $1+x$ leads to a $q (1+1/x)\%$ change in $x$.
For instance, when $x=1$, then a doubling of $x+1$
is the same as a $200\%$ increase of $x$. The $x+1$ changes from $2$ to $4$ and the $x$ from $1$ to $3$.
When $x=100$, then a doubling of $x+1$
is nearly the same as a doubling of $x$. The $x+1$ changes from $101$ to $202$ and the $x$ from $100$ to $201$.


What is x has both negative and positive values with the absolute value less than 1 so that 1+x are all positive?

Transformations like $\log(x+c)$ are seen when $x$ can have zero or negative values. But it changes the interpretation as you see above.
It might be better to solve the problem directly without the transformation. E.g. if the goal is to do some sort of regression and you linearise the function for this with the transformation. Instead, one can also use non-linear least squares regression or a generalized linear model. You don't need to transform data in order to fit a function like $y=a x^b$.
