We have a curve $cx ^ b$. Let's say I sum $x$ 1 through to 52 using that curve and it gives me a number, say 10. I want to increase number 10 by 10% and we know that's 11. I can increase 10% very easily by just multiplying by 1.1. What I want to solve is what I can change in the equation $c* x ^ b$ to make it equal 11 But I only want to change something to do with $b$.
We can assume $x$ can always be limited between 1 to 52
Can someone help me solve this? I would like this solved in the most laymen's terms
I don't know of a nice form for $s=c\sum_{x=1}^{52}x^b$ for $b$ not an integer. There is a nice form for $b$ integer. You need to use a one dimensional root finder. The nice thing is that you have an easy derivative: $\frac {ds}{db}=s \log b$
In your example, the first order approximation is $\Delta b=\dfrac {\Delta s}{\frac {ds}{db}}=\frac 1{10 \log b}$