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.