How to linearize a power function $y=a\cdot x^{b} + c$

Question

I have this power function:

$$y=a\cdot x^{b} + c$$ and I would like to linearise it into the form: $$y=a\cdot x + b$$ I am trying to linearise it because most algorithms for regression are usually done with linear regression. Colleagues of mine told me that I could perhaps put it into the log space to linearise it and do the same statistics as a linear regression. So, I have been trying to put it into the log space but I got stuck. There is ton of help online for the equation in the following form: $$y=a\cdot x^{b} $$ $$log(y) = log(a\cdot x^{b})$$ $$log(y) = b\cdot log(x) + log(a)$$ But I cannot find a way to include the $c$ into this and I need it in my case. So I am stuck here: $$log(y) = log(a\cdot x^{b} + c)$$ Any help would be very much appreciated!

Answer 1

I think the best you can do is to subtract $c$ from $y$ and treat $y-c$ as the quantity you are considering.

If you are trying to linearize it in order to perform least squares optimization on the data, your problem is that you have $3$ parameters instead of $2$. You can't convert it to linear because a line has only two parameters.