Yes, your line of best fit does have to be a line. HOWEVER, since your data looks logarithmic, then a logarithmic transformation to the data would be appropriate to make your data linear.
Note that the model $\displaystyle \displaystyle \begin{align*} y = A + B\log{(x)} \end{align*}$ might be appropriate. If we let $\displaystyle \displaystyle \begin{align*} X = \log{(x)} \end{align*}$, this gives $\displaystyle \displaystyle \begin{align*} y = A + B\,X \end{align*}$, a LINEAR function.
So what you can do is to evaluate $\displaystyle \displaystyle \begin{align*} X \end{align*}$ for each of your observed x values, and then do a least squares linear regression on the two sets $\displaystyle \displaystyle \begin{align*} X \end{align*}$ and $\displaystyle \displaystyle \begin{align*} y \end{align*}$. Your model will then be much more accurate, and then it can be written in terms of $\displaystyle \displaystyle \begin{align*} \log{(x)} \end{align*}$ again.