Its a multiple linear regression model. When I use log both on Y and X's variable, the coefficient should be the elasticity right? But if I am not using the log function, I need to use calculate the elasticity using the coefficient β times the average X over average Y. But although using the same data, both model give different elasticity. Are the average X divide the average Y not accurate for calculating elasticity?

And, yes I am still going through the text book mentioned.