Let's say that I have a product which combines of two elements. I want to set my profit margin based on each element's part. Together both elements sum up to $100$% I would like to charge more commission for smaller part and less for bigger one.
I'm trying to find common exponent using log function in such way that it will sum to 100 + my commission.
For example
$$60+40 = 100 $$
$$60^{x}+40^{x}=115$$
That $15$% commission should be split based on logarithmic function, by applying more weight to smaller part $(40)$.
Also example with 4 parts:
$$40^{x}+40^{x}+15^{x}+5^{x}=120$$
$40$% and $40$% would have the smallest commission and $5$% the largest applied.
Can someone solve both examples for me please?
Rather than trying to guess at a function, I suggest your try to translate your requirements into mathematical terms. As I understand it, in your example, you want $$\begin{align} 60r_1+40r_2&=15\\ 60r_1&<40r_2,\end{align}$$ where $r_1,r_2$ are the commission rates.
Obviously, if the commissions were equal, we would have $$\begin{align} 60r_1&=40r_2=7.5\\r_1&={7.5\over60}=.125\\r_2&={7.5\over40}=.1875\end{align}$$
If you really need the commission on the smaller part to be larger, just choose $r_1$ to be any convenient number less than $12.5\%$ and use the equation $60r_1+40r_2=15$ to determine $r_2.$