Say I have some data, I assume a fit of the form: $\alpha_1 e^{\beta_1 k}$, and I extract some values $\alpha_1>0$ and $\beta_1<0$ for $k=[1,\ldots]$. Say this appears to be a "near perfect" fit for my data, i.e. my data points fall within some $\epsilon$ of $f = \alpha_1 e^{\beta_1 k}$ .
Now, say later I learn that the "true" form of the fit should be: $k \alpha e^{\beta k}$. No longer having access to my data, can I say something about $\alpha_2$ and $\beta_2$ (besides the fact that $|\beta_2|>|\beta_1|$)? I am specifically interested in being able to say something about the ratio: $\frac{\beta_1}{\beta_2}$