How to solve the following multiple linear regression problem?

Question

I have given function values say "$y$" which is dependent on $x$ and $\log(100-x)$, how shall I solve this problem using linear regression - as in I want my model to be like $$y = Ax + B\log(100-x).$$

I could do it by taking $x$ as one variable and $100-x$ as another, but is that wrong?

Is there any other way this could be done using multiple linear regression where $x$ and $\log(100-x)$ are taken as different variables?

Am I committing a mistake by taking $x$ and $\log(100-x)$ as different variables since they are dependent on each other?

How to solve using linear multiple linear regression in R?

Answer 1

If you really want to construct a model such as $y = A x + B \log(100 - x)$ (that is, you know that is how your model SHOULD perform in real life, e.g., based on some physical or biological or chemical arguments), then just do that by treating $x$ and $\log(100 - x)$ as your covariates and perform a linear regression to estimate the coefficients $A$ and $B$.

Answer 2

Linear regression can be used in some non linear regression problems if you define new variables that contains the non linearity. You should do the linear regression $y=A X +B U$ , where $U = log(100-x)$. There is no mistake in doing that, you are searching a linear regression function adding a dimension to the problem.

For example, if you want to adjust the polynomial $Y=AX^2 +BX +C$, you should define a new variable $U = X^2$, and it isn't any mistake on doing that. It is a very common practice while doing regressions.