How to solve equation that contain additive in exponent?

Question

$$ \begin{align} \text{I have equation : } & \left(\frac{y_2+c}{y_0+c}\right)^p=\frac{y_1+c}{y_0+c} & \text{find : } c \text{ when } y_n \text{ is constant} \\ \text{Or in another form : } & \left(\frac{x+a}{x+c}\right)^p=\frac{x+b}{x+c} & \text{find : } x \text{ when } a,b,c,d \text{ is constant} \end{align} $$

Where p is any real number and c is any real number which $x + c > 0$, Can this equation be solved?

On the other hand what I actually did is try to reverse below equation with systems of equation

$$y = ax^b - c$$

Given 3 points of $(x,y)$ that following each other in monotonic fashion should be able to solve this equation, so far I could derived a new equation like so https://www.desmos.com/calculator/ydkxrhdplq

This below image is result I try to made, find $a$ $b$ and $c$ to make this equation pass this 3 points. The problem is I can't find equation for solving $c$ and need to guess it as constant

Or what I try to do is actually not possible?

Answer 1

To approach a solution, try to logarithmize on both sides of the equation, i.e. $$ \log \left(\frac{x+a}{x+c}\right)^p \; = \; \log \frac{x+b}{x+c}$$ In order to simplify the left side: What do you know about $\log y^z$?

Answer 2

After tinkering with the code and convert this equation $x$ and $y$ into vector. It then simplified my equation to make me see its true nature

The complete form of this equation actually is

$$y = a(x - d)^b + c$$

And both $c$ and $d$ is actually the another control point. I don't need to crack out the equation for $c$ at all. I could just complete this equation with the third point

And so it give me this

C# code : https://dotnetfiddle.net/R7mYT9

Desmos Graph of below picture : https://www.desmos.com/calculator/fehckjop8b