Determining a value at a percentage point on an exponential curve

Question

I have a device which is a certain amont on. When the device is 100% on it has the value R. When the device is 0% it has the value 0. The curve of the device is exponential. I am trying to create a formula to represent this.

I am able to do this if I pick specific increments. For example, if I wanted to find the value for every 5% change I could do the following:

I: is the interval, 5 in this example...
P: is the percentage point I am at, e.g., 5, 10, 15, 20, etc...

(R^(100/I))^(P/I)

... the problem is that using this formula I need to know the increments before hand. A percentage could be anything...

How would I write a formula which I could use just R and P as inputs?

Answer 1

A simple exponential curve with $f(0) = 0$ and $f(100) = R$ is $$f(P) = R \left( \frac{e^{P/100}-1}{e-1} \right)$$ where $P \in [0,100]$ is the percentage point.

Answer 2

As you describe it the curve should be $f(P)=Rbe^{aP}$ where $a,b$ are parameters you want to estimate from your data. You can take the log and get $\log (f(P))=\log R + \log b +aP$ If you take measurements of $P, f(P)$ you can fit this to a straight line. If you measure $P$ from $0$ to $1$ and want $f(1)=R,$ then $b=e^{-a}$. Excel has this capability built in. Just list your data in neighboring column, plot and XY plot and add trendline. The equations for doing this are in any numerical analysis test. I like the discussion in Numerical Recipes. Obsolete versions, with code, are free online