Data transformations often facilitate regression analysis, yet many commonly used transformations make hypothesis testing misleading because the results change with the measurement units of the data. We demonstrate both theoretically and using data from a randomized experiment that popular transformations approximating the logarithmic function but accommodating non-positive data can drastically alter regression conclusions when simply changing units, producing findings that are uninformative at best or highly misleading at worst. Our main result characterizes the family of transformations yielding measurement-unit-independent conclusions through an equivalence theorem that links scale-invariant t-statistics, scale-equivariant coefficient estimates, and scale-invariant semi-elasticities with logarithmic and power transformations.