I'm trying to find the coefficients of an exponential curve:
$$y=a^{x-h}+k$$
I've got three pairs of $y$ values, and I know that the $x$ difference between the $y$ values in each pair is 1 unit of $x$, but not the actual $x$ position. I also don't know the $x$ distance between each pair.
The $y$ value pairs are:
$y_1=24$, $y_2=27$
$y_3=48$, $y_4=58$
$y_5=72$, $y_6=104$.
so, for example I know that $x_2 = x_1 + 1$, and $x_4 = x_3 + 1$, and $x_6 = x_5 + 1$, but I don't have the distance between $x_2$ and $x_3$, or $x_4$ and $x_5$
I'm not even sure that this is solvable since I don't have any actual $x$ value samples. (don't use much complex math since uni, so have forgotten alot). I would imagine it involves calculating the slope of of at least one of the pairs, and then finding a way to sub that into the equation, but after that I'm murky.
Another important piece of info is that this is experimental/measured data. I do know that the first pair ($y_1$ and $y_2$) is more accurate than the second pair, and the second pair is more accurate than the third pair given the nature of the measurements, so even an approximation would be useful. For the purposes of this exercise the curve can be assumed to be exponential since other phenomena that are similar are known to also have exponential curves.