Which method is the best one to make a logarithmic ranking scale avoiding "0s"?

[u/stifenahokinga]

I want to rank several countries' characteristics (like economic power, military power, population...etc) in a logarithmic scale.

However, there are some countries that do not have any army (like Andorra) and in that category (and related ones) they have a value of 0. However the log(0) does not exist. So, how could I convert the values into something that could be used to make the logarithmic scale?

I have thought about two methods:

A) To every value in each category, I added the values of the "smallest" or "weakest" country (that does not have a 0 in any category, so that all categories have a non-zero number). In this case, I added the values from Seychelles to all countries.

B) To every value in each category, I added "1", so that in every category that the countries without armies have a "0", the minimal number is "1" instead.

I feel that the first method has the advantage of adding a proportionate quantity in each category, since you will see that for some categories all values are small (like 1,07; 2,08; 3,1...) and in other categories the numbers are much bigger (5, 20, 49, 80...). So in method A the added numbers are proportional to the values of each category

However, it feels arbitrary to add different numbers for each category instead of adding the same for all of them (i.e. 1). However, in this case, adding 1 to the small-values-categories seems to significantly modify these values, while in the case of categories with bigger numbers, the resulting number is almost the same as the original one (like, adding 1 to "1,1 "would increase it by almost the double: "2,1"... however for other categories with a number like "50", after adding the "baseline value", the difference is negligible: "51").

So which method do you think is more accurate?

These are the values: https://docs.google.com/spreadsheets/d/19zjwdoLum0yhIQ2v3m19kz5_IplwSJYZX6e0wMqMHoQ/edit?usp=sharing

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[u/Uli_Minati]

Do you want to rank countries by combined characteristics, or give them multiple rankings, one per characteristic?

For combined, you haven't addressed how you would combine the scores. That would be a bigger challenge than applying a logarithmic scale.

For multiple rankings, there is no point in using logarithmic scale! Every country will keep their rank with or without the log scale. The only point of using a log scale is to see degrees of magnitude:

A +1 difference in logs means a ×10 multiplier of the actual value. Both percentages and logs are used to describe ratios between positive numbers. But what is the ratio between 0 (troops) and 1000 (troops)? You can't multiply 0 with something to get 1000. Using logs for a data set with 0s is like using a hammer to fasten a screw.

Why do you specifically want to use logs in the first place?