How to properly get log or exp parameters given a sample table of a exp/log curve?

Question

What is the correct process to get the correct formula of conversion between X and Y (in both directions) where

        Y       X
        1       0
        0.5     -6
        0.25    -12
        0.125   -18

I would say, intuitively, that

X is 6 times a log2 of Y, but I learnt to avoid using intuition, if I can't get something more formal, I would not trust my intuition.

Also Y will be necessarily > 0

Answer 1

You are correct. Note that
$$\begin{array}{c|c|} & \text{Y} & \text{X} \\ & \big(\frac{1}{2}\big)^{0} & 0\cdot(-6) \\ & \big(\frac{1}{2}\big)^{1} & 1\cdot(-6) \\ & \big(\frac{1}{2}\big)^{2} & 2\cdot(-6) \\ & \big(\frac{1}{2}\big)^{3} & 3\cdot(-6) \end{array}$$

So we have $$Y=\big(\frac{1}{2})^{-\frac{X}{6}}=2^{\frac{X}{6}}$$ and it follows that $X=6\log_2{Y}$ for $Y>0$.

You can try a solution of the form $Y=a^{bX}$, substitute $(Y,X)=(1,0)$, $(Y,X)=(0.5,-6)$ and solve for $a$ and $b$. Then check it satisfies the remaining coordinates. Generally it is not easy to find a "nice" solution.