Suppose I have two points in $\mathbb{R}^2$ and I wish to find values of parameters $a$ and $b$ such that I obtain the power law $y=ax^b$ which goes through the two given points. I can solve the nonlinear system$$\begin{eqnarray}y_1 &=& ax_1^b\\y_2 &=& ax_2^b \end{eqnarray}$$
or I can take the log of both sides and obtain$$\begin{eqnarray}\ln(y_1) &=& \ln(a)+b\cdot\ln(x_1)\\\ln(y_2) &=& \ln(a)+b\cdot\ln(x_2) \end{eqnarray}$$
a linear system which can be easily solved by using a direct solver for example and then I can obtain $a$ and $b$ easily. I can do the same for an exponential ($y = ae^{bx}$) and even a Gaussian $\left(y = ae^{-\frac{1}{2}\left(\frac{x-b}{c}\right)^2}\right)$.
My first question is, is this approach possible for the Lorentzian function ($y=\frac{a}{b+(x-c)^2}$)? What kind of a transformation would I do?
Does this technique have a name? Is there a general approach to this? Given an arbitrary function, is there a way to determine if such a method is possible or not? If yes, how do we determine the appropriate transformation?