I'm a programmer working to write test software. Currently estimates the values it needs with by testing with a brute force algorithm. I'm trying to improve the math behind the software so that I can calculate the solution(s) instead. I seem to have come across an equation that is beyond my ability.
$$(Bx_1 + 1)^{y_2}=(Bx_2 + 1)^{y_1}$$
My goal is to have $B$ as a function of everything else, or have an algorithm solve for it. I feel like it should be possible, but I not even sure how to begin.
Okay so I came up with a that allows me to approximate the answer if $\frac {y_2}{y_1}$ is rational (which it will be in my case because I have limited precision).
I can re-express the original equation as $$(Bx_1+1)^{\frac {y_2}{y_1}}=Bx_2+1$$
If $\frac {y_2}{y_1}$ rational I can change it to $\frac ND$ where $N$ and $D$ are integers. Substituting this back in and redistributing the fraction I get $$(Bx_1+1)^N=(Bx_2+1)^D$$
Here is the fun part, here I can do a binomial expansion on each side, this will give me one high-order polynomial that I can approximate the answer to. It's very messy but should get the job done.