Why is there more weight on smaller y values in transformed linear regression as compared to least squares regression for exponential models?
This is a typical problem with functions which vary very fast. In fact, using the logarithm is the same as minimizin the relative error instead of the absolute arror.
Consider $$\Delta_i=\log(\hat y_i)-\log( y_i)=\log\left(\frac{\hat y_i }{y_i } \right)=\log\left(\frac{\hat y_i-y_i+y_i }{y_i } \right)$$ $$\Delta_i=\log\left(1+\frac{\hat y_i-y_i }{y_i } \right)\sim \frac{\hat y_i-y_i }{y_i } $$
I remember having worked this problem for you and as you saw the parameters changed a lot.