Where does this equation come from and the reason behind it to a model exponential function?

Question

To model data based on the exponential equation

$$ y=Ae^{Bx} $$

On the website https://mathworld.wolfram.com/LeastSquaresFittingExponential.html

In equation (5)

$$ \sum_{i=1}^{n}{y_i\left(\ln{\left(y_i\right)}-a-{bx}_i\right)^2} \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space \space\space\space\space\space\space\space\space\space\space\space (5) $$

where

$$ \ln{\left(A\right)}=a $$

$$ B=b $$

It says this sum should be minimized. It works better than the linearizing the data and doing linear regression for the equation below.

$$ \ln(y)=\ln(A)+Bx $$

I know that this works better but what is the justification for the use of equation (5) or how was it derived?

Answer 1

Answering your last question, I showed that, if you express the residue as $$r_i=\log(\hat y_i)-\log( y_i)$$ assuming small errors, you have $$r_i \sim \frac{\hat y_i-y_i }{y_i }$$ which explains that the linearization leads to the minimization of the sum of the squares of relative errors.

So, if now, you express the residue as $$r_i=y_i \Big[\log(\hat y_i)-\log( y_i)\Big]$$ with the same assumptions, you have $$r_i \sim y_i \times \frac{\hat y_i-y_i }{y_i }=\hat y_i-y_i$$ which makes that this weighting factor leads to something extremely close to the minimization of the sum of the squares of absolute errors.