I'd like to create pricing curve that's based upon a reverse exponential function. I know the starting point and ending point, but don't know how to create the curve in between.
For example, say for 100 units we have a price of 50 p/unit, and for 500 we have 25 p/unit. How can I create the range of values in between?
So our exponential function will be for the form: $P = C_1*C_2^u$
where P = price, u = units, and $C_1, C_2$ are two constants we are attempting to determine.
To solve this consider the two cases you were given:
$$ 50 = C_1*(C_2)^{-100} $$
$$ 25 = C_2*(c_2)^{-500} $$
We now take the $\log_{10} $ of both sides
$$ \log_{10}(50) = \log_{10}(C_1*(C_2)^{-100}) $$
$$ \log_{10}(25) = \log_{10}(C_1*(C_2)^{-500}) $$
This we expand the products using the log product to sum rule:
$$ \log_{10}(50) = \log_{10}(C_1) + \log_{10}((C_2)^{-100}) $$
$$ \log_{10}(25) = \log_{10}(C_1) + \log_{10}((C_2)^{-500}) $$
We now simplify the exponents using the log power to product rule
$$ \log_{10}(50) = \log_{10}(C_1) - 100*\log_{10}((C_2)) $$
$$ \log_{10}(25) = \log_{10}(C_1) - 500*\log_{10}((C_2)) $$
subtract the top equation by the bottom
$$ \log_{10}(50) - \log_{10}(25) = 400*\log_{10}((C_2)) $$
$$ \log_{10}(25) = \log_{10}(C_1) - 500*\log_{10}((C_2)) $$
Therefore:
$$ 10^{\frac{\log_{10}(\frac{50}{25} = 2)}{400}} = C_2$$
Now substitute this value of $C_2$ back into the second equation and solve for $C_1$