Transform exponential expression into log / find solution to: $y = (a - x_1)^\gamma + (a - x_2)^\gamma$

Question

I need to find the solution to the expression $y = (a - x_1)^\gamma + (a - x_2)^\gamma$ for $a$, but have problems with exponential, where $\gamma $ can be greater than 0 and smaller than 1 or greater than 1: (1) $0 < \gamma < 1$ and (2) $\gamma > 1$. I think $(a - x_1)^\gamma$ has to be to transformed into logarithm, but I can't find the way to solve it. I would appreciate any hint or help.

Answer 1

If I can assume $a>x_1>x_2$, which is ok if $x_1-x_2<y^{1/\gamma}$, then $a$ is between $x_1$ and $y^{1/\gamma}+x_2$. You can then try the Bisection method to home in on the answer.