Creating an exponential scale

Question

Good morning,

I am trying to create an exponential scale for attributing values in a scoring model.

Here is the function I was thinking of using:

y = z^x

Where:

y = Score X = Risk assessment value that goes from 1 to 10

Now here is my thing. I need to make sure that the minimum value of Y is 25 and its maximum value is 300. I don't care what the value of z is. X has values between 1 and 10 I need to make a function so that Y fits my minimum and maximum requirements:

Min --> 25 = z^1 Max --> 300 = z^10

Y can take any value between 25 and 300 when the x is comprised between 2 and 9.

I guess I should create a systems of equations?

Thanks, Bernardo

Answer 1

Exactly what you're asking for is not possible. If $25 = z^1$, this implies $z=25$. But this does not satisfy $300 = z^{10}$.

What you are looking for (I guess) is an equation of the form $$y = C \cdot z^x$$ with the conditions

$$25 = C \cdot z \leftrightarrow C=\frac{25}{z}$$

and

$$300 = C \cdot z^{10} \leftrightarrow C=\frac{300}{z^{10}}$$

Setting the two equations equal to each other we get:

$$\frac{25}{z} = \frac{300}{z^{10}} \leftrightarrow z^9 = \frac{300}{25}$$

with the solution $z=(\frac{300}{25})^{\frac{1}{9}} \approx 1.32$. Substitute the value for $z$ into one of the equations above to obtain the value of $C \approx 19$.