
Least squares problem
I'm having trouble to understand the consign of the exercise. It says "Approximate the data of the following table with a model of the form $\displaystyle f(x)$~$\displaystyle ae^{bx}$ in the meaning of least squares for the function $\displaystyle ln(f(x))$.
$\displaystyle \begin{array}{ccccc} x & 1 & 0 & 1 & 2\\ \hline y & 8.1 & 3 & 1.1 & 0.5\end{array}$"
My approach : I believe they ask me to approximate the data in the table, with a function of the form $\displaystyle f(x)=ae^{bx}$, so I wonder what the $\displaystyle ln(f(x))$ is about. So I didn't considered it. I made the least squares, that is $\displaystyle f(a,b)=\sum_{i=1}^2 (ae^{bx}y_i)^2$. I did the partial derivative with respect to a and equaling it to 0 I got the following equation : $\displaystyle ae^{bx}c=0$, where $\displaystyle c$ is $\displaystyle \sum_{i=1}^2 y_i$.
I did the same with the partial derivative with respect to b, except that I had to suppose $\displaystyle a$ and $\displaystyle b$ different from $\displaystyle 0$ to make some division. And I got exactly the same equation as I got for the first partial derivative. Therefore the system has no solution, which is not normal considering the problem given. It means I made an error, or I misinterpreted the problem. Can you help me?