How to back transform when there is transformation in IV and DV

Question

I understand an oversimplified way to backtransform data is to reverse the operation. Note: I realise this isn't necessarily ideal.

e.g squared data gets back transformed via square root.

However, what happens in the case where the dependent variable has been transformed and one of the independent variables in a multiple regression has been transformed?

Keeping it basic, is it as straightforward as back transforming both?

Or could you just back transform the DV and leave the IV transformed, but you'd need to note to yourself or anyone using your model that the particular IV is backtransformed in the model, and they may need to transform this variable prior to putting it into the model.

e.g.

Carparks = 5 + a1(x) + a2(x)

Carparks, however, is really the log of car parks and a2 is really the log of a2. For anyone using the model I could say 'you'll need to log transform your a2 data before putting it in to this model'.

Answer 1

I agree with the comment from @StatsStudent about the independent variables and would argue you should stay in the log scale for the dependent variable as well.

There is a long history of having log-transformed variables on both sides of a regression in economics, where the regression coefficients are then known as "elasticities." Your coefficient for $\log(Carparks)$ as a function of $\log(a_2)$ represents the percentage change in Carparks per percentage change in $a_2$. That will often be easier for others to understand than any back-transformed results would be.