Linear Rescaling to Accurately Interpret Logarithms

TL;DR

This paper proposes using base-(1+p) logarithms for more accurate and interpretable proportional changes in statistical analysis, improving precision over traditional natural logs especially for larger p values.

Contribution

It introduces a novel approach of using base-(1+p) logarithms to interpret proportional changes exactly, reducing approximation errors in statistical models.

Findings

Exact interpretation of proportional changes with base-(1+p) logs

Reduced error compared to traditional log(1+X) transformations

Improved approximation quality in statistical analysis

Abstract

The standard approximation of a natural logarithm in statistical analysis interprets a linear change of \(p\) in \(\ln(X)\) as a \((1+p)\) proportional change in \(X\), which is only accurate for small values of \(p\). I suggest base-\((1+p)\) logarithms, where \(p\) is chosen ahead of time. A one-unit change in \(\log_{1+p}(X)\) is exactly equivalent to a \((1+p)\) proportional change in \(X\). This avoids an approximation applied too broadly, makes exact interpretation easier and less error-prone, improves approximation quality when approximations are used, makes the change of interest a one-log-unit change like other regression variables, and reduces error from the use of \(\log(1+X)\).