Solve for c , $y = x + c \big( \frac{mx}{c} + s \big)^a$

Question

I get this equation

$y = x + c \big( \frac{mx}{c} + s \big)^a$

how can I get the $c$ or $m$ ?

I try with $\ln$

$\ln\big(\frac{y-x}{c}\big) = a \ln \big( \frac{mx}{c} + s\big)$

and now ?

Answer 1

The m is easy:

y = x + c(m*x/c+s)^a
((y-x)/c)^(1/a) = (m*x/c+s)
c*(((y-x)/c)^(1/a) - s)/x = m

C is a little tougher, you're actually not going to get a closed form for it. I say that because I plugged it into Mathematica, who said they couldn't, and they're REALLY clever about that stuff.

Answer 2

Here is how you solve for $m$:

\begin{align*} \frac{y-x}{c}=\left(\frac{mx}{c}+s\right)^{a}\\ \left(\frac{y-x}{c}\right)^{\frac{1}{a}}=\left(\frac{mx}{c}+s\right)\\ \left(\frac{y-x}{c}\right)^{\frac{1}{a}}-s=\frac{mx}{c}\\ \end{align*}

Hence \begin{align*} m=\frac{c}{x}\left(\left(\frac{y-x}{c}\right)^{\frac{1}{a}}-s\right) \end{align*}

I think to solve $c$, you have to use numerical method since it is not explicit.