0

I am trying to come up with a formula so I can get the value of $X$, with whatever $Y$ I put in. A few example values are listed down below

\begin{matrix} Y & X \\ 1 & 0.9 \\ 10 & 0.5 \\ 100 & 0.3 \\ 1000 & 0.2 \\ 10000 & 0.15 \\ 100000 & 0.125 \end{matrix}

So if I were to for example input $Y$ as $1000$, $X$ would be $0.2$
So if I were to for example input $Y$ as $100$, $X$ would be $0.3$

Now lets say I wanted to input Y as $758$ or $29$ what would $X$ be?

This formula is what I am trying to figure out.

Help would be very much appreciated!

Thanks and kind regards,
- Nick

MonkeyKing
  • 3,178
Nick van Wersch
  • 3

1 Answers1

0

I guessed out one which I guess is what you intended: $$X = 0.8 \cdot 0.5^{\log_{10} Y} + 0.1$$

The idea is that if you subtract $0.1$ from $X$, say you get $X'$, then $X'$ is halved every time as $Y$ gets an extra $0$.

But this question is ill-posed since the above formula is not the only one, you can even have infinitely many continuous formulae. This one just looks "nice". Here is a "not nice" one: let $X$ equal value you given in the table with specific $Y$ mentioned in the table, and let $X=0$ for all other $Y$'s.

MonkeyKing
  • 3,178
  • Just wondering, is it possible to achieve this while not using log? – Nick van Wersch May 12 '15 at 01:30
  • It's already "very beautiful", I doubt there will be any formula more beautiful than this one. OK, another formula that fits your condition: connect adjacent points with straight line. No logarithm, all derivative exist except 4 points. But it is not a smooth curve. This is why your question is ill-posed, there is no answer. – MonkeyKing May 12 '15 at 01:35