Exponential Function with start and end point

Question

I have the following situation.

I have an start point of 40 degrees temperature and endpoint of 69 degrees.

Now i want to normalize all values in this range into an skala from 1-15.

This should happen exponentally. I already have a formular based on an linear approach, but i want the number between 1-15 rise exponentally the more the temperatures comes to the endpoint 69.

Example: 40 degrees = 0P
         51 degrees = 4P
         61 degrees = 12P

Could you point me out to the right direction. I dont know why but i am having a mind blockade at the moment ^^

Thank you in advance!

Answer 1

First normalize by subtracting by $39$... This assures we can work with an Exponential. We also subtract $1$ from the Scala... This gives us $x^0 = 1$ as we need, so interrogating for only one value is needed

$1$ degrees = $ 0$
$30$ degrees = $ 14$
This implies that $x^{14} = 30 \implies x \approx 1.275$
You will now need to get your data points from that curve, not the other way around. In general, you have the function $1.275^{x-1} + 39 = y$, where $y$ is the temperature in degrees and $x$ is the Scala value