Exponent bound for a binomial power equation ($a{x}^n-b{y}^n=c\ne 0,\max (x,y)>1$)

For fixed positive $a,b$ and fixed nonzero $c$, with $x\ne y$, put $Y=\max (x,y)$ and take $\Lambda =\log (a/b)+n\log (x/y)$, swapping the variables and coefficients if necessary. The equation gives $\log |\Lambda |\le -n\log Y+O(1)$ when the larger-variable power is sufficiently large. A two-logarithm lower bound gives $\log |\Lambda |\ge -C(a,b)\log Y\log (2n)$. Hence $n\le C\log (2n)+O(1)$, so $n$ is bounded. The cases $x=y$ and a bounded larger-variable power are elementary. For each remaining exponent, the Thue theorem for a binomial form supplies finitely many pairs. The constant-power exception $x=y=1$, $a-b=c$, must be removed.