Logging an x-variable

Summary

• In a linear model where one or more explanatory variables are logged we have

$$β_j= \frac{∂E\left( y|x \right)}{∂log x_j}= \frac{∂E\left( y|x \right)}{∂x_j}x_j$$

• for all logged explanatory variables. $β_j$ /100 measures, approximately, the unit increase in $E\left( x \right)$ if we increase $x_j$ by 1% keeping the other explanatory variables constant.