Find an exponential formula with two inputs and one output

Question

I'm starting with a pretty much blank understanding of calculus and trigonometry. I'm not even sure either one is necessary here. I'm familiar with basic Algebra. I'm interested in learning more about these subjects, but most of the articles I've come across start at a level that is too far over my head for me to understand it. I assume an understanding of either Interpolation or Least Squares(perhaps both) would help me to solve this easily, but my existing knowledge isn't enough for me to decipher those texts.

If you could point me toward a resource that could help me build to an understanding of either of those subjects, I would greatly appreciate it(assuming I'm correct in my assumption above). Even better though, if you could provide an answer that would satisfy the below situation, I would be ecstatic. :D

Basic premise: A product of the combination of base(b) and work(w) results in a predictable gain(g).

Here is a small subset of the data I have(b:w, g):

    111:100,  1
    863439:50,  3137
    7000:1000,  231768
    200000:4000,  11845572
    35000:6019,  18190000

I've intentionally included a wide range within this subset. As you can see, increasing the magnitude of work(w) seems to have a much larger effect than increasing the magnitude of base(b).

Answer 1

Input data $$ \begin{array}{rrr} x & y & z \\\hline 111 & 100 & 1 \\ 863439 & 50 & 3137 \\ 7000 & 1000 & 231768 \\ 200000 & 4000 & 11845572 \\ 35000 & 6019 & 18190000 \\ \end{array} $$

Functional form

$$ z(x,y) = a x^{b} y^{c} $$

Least squares problem

$$ \left[ \begin{array}{ccc} a \\ b \\ c \\ \end{array} \right]_{LS} = \left\{ \left[ \begin{array}{ccc} a \\ b \\ c \\ \end{array} \right] \in\mathbb{C}^{3} \colon \, \sum_{k=1}^{5} \left( z_{k} = a x_{k}^{b} y_{k}^{c} \right)^{2} \text{ is minimized} \right\} $$

Least squares solution

$$ \left[ \begin{array}{ccc} a \\ b \\ c \\ \end{array} \right]_{LS} = \left[ \begin{array}{l} 0.013405 \\ 0.249977 \\ 2.11573 \\ \end{array} \right] $$

Error vector $$ r = z - a_{LS} x^{b_{LS}} y^{c_{LS}} = \left[ \begin{array}{c} -740.322 \\ 1531 \\ -40902.1 \\ 4742.05 \\ 8668.41 \\ \end{array} \right] $$

Total error $r^{2} = 1.77\times 10^9$

The plot below shows $r^{2}$ as a function of $b$ and $c$ with $a$ fixed at $a_{LS}$.