How to write an equation where both independent variables and dependent variables are log transformed in a multiple regression?

Question

How to write the multiple regression model when both the dependent variable and independent variables are log-transformed?

I know that without any log transformation the linear regression model would be written as

But now I have transformed both my dependent variables and independent variable with log. So is correct to write as

Or since I am transforming both sides of question so can I write it as

Answer 1

I am afraid that all your transformed forms are false.

Starting from

$$y=\beta_0+\beta_1(x_1)+\beta_2(x_2)+...$$

you can write :$$\ln(y)=\ln\bigg(\beta_0+\beta_1(x_1)+\beta_2(x_2)+...\bigg)$$ This is not equivalent to $\beta_0+\beta_1(\ln(x_1))+\beta_2(\ln(x_2))+...$

Also you can change of variables : $$X_1=e^{x_1}\quad;\quad X_2=e^{x_2}\quad,...$$ $$y=\beta_0+\beta_1\ln(X_1)+\beta_2\ln(X_2)+...$$ And with the change of : $$Y=e^y$$ $$\ln(Y)=\beta_0+\beta_1\ln(X_1)+\beta_2\ln(X_2)+...$$ Sorry if I misunderstood your question.