How can I find the value of the coefficients of an inverse exponential graph? If I have a set of discrete data points and I want to find the line of best fit of the form $$y=Ae^{-Bx}$$
then I can do the following:
$$\ln(y)=-Bx+\ln(A)$$
Therefore, if I plot $(x,\ln(y)$), the gradient is the $-B$ coefficient of my line of best fit and my $y$-intercept is $\ln(A)$ of my line of best fit.
Note: The above method, finds the values of the coefficient of my line of best fit using all data points, and not just choosing two and solving a set of equations.

Now, if I have a set of data points and I want to fit a function of the following form:
$$y=A(e^{-Bx}-1)+C$$
how can I find the value of the coefficients? The method above would not work for this scenario. Instead how could I identify the value of the coefficients $A, B$ and $C$?
Note: ${A,B,C \in \mathbb R }$ and $x$ and $y$ can be either positive or negative.
 A: I presume you are already aware that by doing a linear regression on $\log y$ you are minimizing
the relative error of $y$ , i.e. $\Delta y/y$ and not just $\Delta y$.
That said, in linear regression the line is always passing through the barycenter of the cloud of points $\left( {\overline x ,\,\overline y } \right)$.
So I would suggest that you operate in this way
$$
\eqalign{
  & y = A\left( {e^{\, - Bx}  - 1} \right) + C = Ae^{\, - Bx}  + D  \cr 
  & \quad  \Downarrow   \cr 
  & \left( {y - \bar y} \right)
 = \left( {Ae^{\, - B\,\bar x} } \right)e^{\, - B\left( {x - \bar x} \right)}  + \left( {D - \bar y} \right)  \cr 
  & \quad  \Downarrow   \cr 
  & \left\{ \matrix{
  D - \bar y \hfill \cr 
  \Delta y = A'e^{\, - B\Delta x}  \hfill \cr}  \right. \cr} 
$$
A: Performing a linear best fit to the log of the data works if you assume that the measurements' noise are Gaussian distributed after taking the log. While there may be some trick to reduce the second case to a linear fit, it is probably better to assume that the noise is Gaussian in the original form and perform a non linear regression, based on a maximum likelihood estimation.
