# Struggling with data transformations that can produce negative values

My question: Which data transformations of positive values produces negative values on the transformed scale?

The context: I'm looking at a recently published paper where there is a figure that reports that the x-axis is in meters on a log-10 transformed scale. I'm very interested what that actual range of the raw distance data was. The original values should have varied from 0 to perhaps 5000 m (though they don't indicate the actual distances in the article). However, the values on the scale range from -2.0 to 0.5 (see figure reproduced below), and neither log nor natural log transformation can produce negative values. Log values should vary from 0 to ~3.5 and natural log from 0 to ~8.5. They used an arcsine transformation for percentage data elsewhere in the paper, but that also cannot produce negative values (and cannot deal with values great than 1, I believe). Square root transformations obviously cannot produce negative values either. • Try taking the natural logarithm of a number in (0,1). – soakley May 4 '13 at 13:20

I see no need to answer the general question, as I think your post just arises from a misunderstanding of logarithms.

The logarithm of any positive number to any base is negative when that number is below 1. log10(0.01) is -2, for example. By eye the smallest logarithm on the graph is about -1.7, which would be about 0.02 metres.

I am not impressed by the graph. It is easy enough in decent software to use a log scale and also show values in the original units. As your question shows, that will be clearer to many statistics users.

[LATER] Matt Krause seems very likely to be right about miles, not metres. What meaning is there in saying that agriculture is 20 mm away? I suppose you could be standing at the farm gate....

• I think the OP may be mistaking the domain of log(x) i.e. x>0, with the range of log(x) i.e. $log(x) \in (\infty, \infty)$ – conjectures May 4 '13 at 15:25
• @conjectures You need a negative sign in there too. I think we are all making similar comments in different style. – Nick Cox May 4 '13 at 22:28

I'm afraid you're confused: Logs can easily produce negative values! For example, $log_{10}(0.01) = -2$, since $10^{-2}=0.01$. You cannot, however, take the log of zeros or negative values,as log goes to negative infinity, which is what you might be thinking of.

That said, it looks like your graph goes from about 0.01 to around 0.5. Are you sure [m] is means meters and not miles here?