I want to interpolate with the function $$f(x) = a\ln(x+b)+c$$ That is, I assume some sort of logarithmic relationship, but there might be an offset.
I assume that I need 3 datapoints, as there are 3 parameters $a,b,c$. Let's say $f(x_1) = y_1; f(x_2) = y_2; f(x_3) = y_3$. Now I want to determine $a$, $b$ and $c$.
$$ y_1 = a\ln(x_1+b)+c; y_2 = a\ln(x_2+b)+c; y_3 = a\ln(x_3+b)+c $$ Thus $$ a\ln(x_1+b) - y_1 = a\ln(x_2+b) - y_2 = a\ln(x_3+b) - y_3 $$ Thus $$ \frac{\ln(x_1+b) - \ln(x_2+b)}{y_1 - y_2} = \frac{\ln(x_1+b) - \ln(x_3+b)}{ y_1 - y_3} $$ Thus $$ (x_1+b)^{y_3-y_2} \cdot (x_2+b)^{y_1-y_3} \cdot (x_3+b)^{y_2-y_1} = 1$$ Now I am stuck and don't know how to solve this for $b$.
Is my derivation correct?
Is there a solution in closed form?
If not, is it possible to make some assumptions regarding $x_1$ $x_2$ and $x_3$ so that there is a solution in closed form?
Would it be easier, if $b$ was assumed known? (We would need only 2 datapoints in this case.)
Your derivation is correct, but not going in the right direction. You should view your three data points as giving you three equations in the three unknowns $ a,b,c$, which you are trying to evaluate. You are correct that you won't get there analytically because of $b$. If you fix $b$, it is easy. You have $y_1-a\log(x_1+b)=y_2-a\log(x_2+b)$, which you can solve for $a$, then get $c$. If you do want to fit for $b$, I would pretend $b$ is fixed, follow the above to get $a,c$, and check the error at your third point. Feed this error function to your favorite root finder to get $b$