Advantage of using the natural logarithm
Logarithmic transformations are often done to base 10 or to base e (natural logarithm). An advantage of the latter is that in the case of small numbers one can fairly accurately exponentiate without the use of computers or tables. Say, you regressed log transformed data (base 10 and base e separately) on some other variables, and the RSD (residual standard deviation) is of the order of –0.14 to 0.14. The table below shows the relationship between RSD and its antilogarithm.
| RSD | antilogarithm of | RSD | antilogarithm of | |||
| log10 | loge | log10 | loge | |||
| 0.01 | 1.023 | 1.010 | -0.01 | 0.977 | 0.990 | |
| 0.02 | 1.047 | 1.020 | -0.02 | 0.954 | 0.980 | |
| 0.03 | 1.072 | 1.030 | -0.03 | 0.933 | 0.933 | |
| 0.04 | 1.096 | 1.041 | -0.04 | 0.912 | 0.961 | |
| 0.05 | 1.122 | 1.051 | -0.05 | 0.891 | 0.951 | |
| 0.06 | 1.148 | 1.062 | -0.06 | 0.871 | 0.942 | |
| 0.07 | 1.174 | 1.073 | -0.07 | 0.851 | 0.932 | |
| 0.08 | 1.202 | 1.083 | -0.08 | 0.832 | 0.923 | |
| 0.09 | 1.230 | 1.094 | -0.09 | 0.813 | 0.914 | |
| 0.10 | 1.259 | 1.105 | -0.10 | 0.794 | 0.904 | |
| 0.11 | 1.288 | 1.116 | -0.11 | 0.776 | 0.896 | |
| 0.12 | 1.318 | 1.127 | -0.12 | 0.759 | 0.887 | |
| 0.13 | 1.349 | 1.139 | -0.13 | 0.741 | 0.878 | |
| 0.14 | 1.380 | 1.150 | -0.14 | 0.724 | 0.869 | |
Conclusions
- Only with the natural logarithm can one fairly accurately estimate the proportional scatter by heart. For example, an RSD = 0.05 implies a scatter of about 5% (one times the standard deviation comes to 1.051 times (5.1% more), or to 0.951 times the predicted number (4.9% less).
- If the scatter of log-transformed data is normally distributed, this is not the case for the untransformed data. Example: if the RSD of loge(Y) = 0.10, then one RSD above predicted comes to 110.5% (+10.5%), and one RSD below predicted comes to 90.4% (-9.6%) of predicted.