log(x * constant) transformation I am a conservation biologist and I am creating a new index to classify species extinction risk. The index ranges from one to zero, with scores closer to one representing a higher extinction risk. Due to the distribution of the data, the calculated values are always very skewed: few species have a value close to one and most species have a value close to zero.
This pattern seems to be true, as a few species do have everything going wrong for them, which results in a disproportionate extinction risk, but this does not mean other species are not at risk as well. Moreover, this skewed distribution makes it hard to visualize any geographical or phylogenetic pattern in extinction risk.
To fix such issues, I decided to transform the data with the classic log(x+1) transformation, but that did not help a lot. I tried different transformations and what gave me the most satisfactory results was a log(x*constant) transformation, in which the constant is an arbitrarily large number, such as 10^100.
Trying to anticipate reviewers' objections, I looked for other examples of such transformation being used in my field, but could not find any.
My questions are:

*

*Is this transformation something very unusual, or is it employed in any other fields?

*Is there any reason a reviewer could object to its use?

*Is there a more usual transformation that would yield similar results?

Edit, for clarification:
I am not doing a regression. I calculate an index based on several threats, such as deforestation and climate change and this index is the final product. If I want to display it on a map, for example, the skewness in the index will be a problem. The map will show only one region with a very high value (red) and the rest of the world with low values (blue), and I won't see any meaningful patterns there. Using log(x+1) shows a very similar map. Using log(x*constant) gives a more nuanced map, with different regions colored in different shades of red/blue, which is much more useful.

As noticed in the comments,
$$
\log(x \times c) = \log x + \log c
$$
by the properties of the logarithms. So what you are doing is you shift the data by a constant. It doesn't change anything about the interpretability of the results (on the log scale), their numerical values are just shifted.
If this improves the readability of the plots, it's just because apparently internals of the plotting software you use did not figure out by itself how to scale it if you used the raw data.
If you get different results of regression analysis before vs after the transformation, it seems like you are running into numerical issues with the software you are using. Mathematically speaking, this transformation should only change the intercept of the linear regression model since it is additive.
