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In economics, it is common to use dimensioned variables, e.g. earn- ings (measured in dollars per year), as arguments in the logarithmic function. This is conceptually problematic because a logarithmic func- tion can only take dimensionless quantities as its argument. One way to avoid this conceptual error is to rewrite commonly used logarithmic regressions using an arbitrarily chosen reference unit so that ratios of dimensioned quantities are used in logarithmic functions. With the addition of a zero conditional mean assumption about the reference unit to the standard list of assumptions about asymptotic properties of ordinary least squares estimators, such a reformulated model can ensure consistent estimation of elasticities and semi-elasticities with- out relying on conceptually problematic mathematical operations.
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Basu, Deepankar, "Dimensional Analysis and Logarithmic Transformations in Applied Econometrics" (2022). Economics Department Working Paper Series. 339.