Let's say we have a model like:
$$log(y) = \beta_{1} + \beta_{2} \cdot log(X_{2})+\beta_{3} \cdot X_{3} + u$$
After carrying an OLS, we are asked which independent variable has a higher impact on the dependent variable, for which we are interested in compute the BETA coefficients.
I know that normally (given a model like $y = \beta_{1} + \beta_{2} \cdot X_{2}+\beta_{3} \cdot X_{3} + u$) we would just compute:
$$BETA_{2} = \widehat{\beta}_{2} \cdot (S_{x_{2}}/S_{y})$$ $$BETA_{3} = \widehat{\beta}_{3} \cdot (S_{x_{3}}/S_{y})$$
interpreting those BETA as standardized coefficients; the higher they are, the higher the impact of the independent variable over the dependent variable.
But, dealing with log-log and log-level, I suspect that the comparison is not as straighforward, since, when interpreting semi-elasticities, we have to take into account that increasing $X_{3}$ by one unit is associated with a $(100 \cdot \widehat{\beta}_{3})$% increase in $y$. Then, in order to standardize the coefficients and interpreting them, would
$$BETA_{2} = \widehat{\beta}_{2} \cdot (S_{x_{2}}/S_{y})$$ $$BETA_{3} = 100 \cdot \widehat{\beta}_{3} \cdot (S_{x_{3}}/S_{y})$$
be correct?