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I have read in many different studies now that take the natural logarithm of one plus x. For example, in econometrics many studies use the natural logarithm of one plus the total assets.

I do not understand why, because assets are so big anyways that adding one would not make a difference, or?

For example:

Natural Logarithm of one plus Assets(1,000,000) would be NL(1,000,001).

Limps
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    This is just a guess. If one of the numbers in the data set is equal to $0$ then $\log(0)$ becomes undefined, or if you prefer, $-\infty$. To keep your quantities as real numbers you add $+1$ to them. In addition, $\log(1+x) \approx x$ if $x$ is a small quantity (the linear approximation). – Nicolas Bourbaki Dec 30 '22 at 08:09
  • Thank you so much, so it is a rather "technical" cue and I can simply add one by log(1+x) without worrying that I skew the data? – Limps Dec 30 '22 at 08:28
  • You are definitely changing the distribution of your data, however, since you know what function was applied to your data, you can always go back to the original by reversing the process, so changing the data distribution is not a problem. In fact, it might actually be a good thing, since after taking logs your data will range from $-\infty$ to $\infty$, so in particular, there might be a chance that the transformed data will be normally distributed (which is probably what you want). – Nicolas Bourbaki Dec 30 '22 at 08:49
  • Whenever this transformation makes sense -- because values are $\ge 0$ -- the usual effect will be to reduce skewness in the data. – Nick Cox Dec 30 '22 at 11:35
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    @NicolasBourbaki Negative results are impossible with this translation. The point is to map values 0 upwards to 0 upwards. – Nick Cox Dec 30 '22 at 11:56
  • This transformation is arbitrary and can really move estimates around depending on what you add to zero. More on this, with alternatives, here. – dimitriy Dec 30 '22 at 15:05
  • @dimitriy You're referring to $\log (x + c), c > 0, x \ge 0$, but nothing in the question is about that. – Nick Cox Jan 03 '23 at 16:12
  • @NickCox Isn’t the question about $\ln (x + c)$ for $c=1$? I am making a claim that adding a penny or a dollar can have a noticeable impact on the coefficients and the choice seems arbitrary. – dimitriy Jan 03 '23 at 18:45
  • That is so and what Mullahy and Norton are on about in the paper you cite. I think the proof of the pudding is in the eating and to be sure in practice, one needs to think about units. – Nick Cox Jan 03 '23 at 19:05

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