t^r - t = c; subs(t= exp(y), y= x*ln(b), r=ln(a)/ln(b), %): simplify(%) assuming 0<x, 0<b; x x a - b = c For the example t^r - t - c is (almost) a line at the solution.

To: math-fun@mailman.xmission.com Subject: [math-fun] a^x-b^x=c, solve for x ?

I can solve this equation with Newton's iterations, but I was curious if there might be an elegant closed form solution.

I tried fiddling with hyperbolic trig functions, but nothing seemed any simpler.

real a>b>0,c>0

E.g., a ~ 50, b ~ 49, c ~ 2 => x ~ 1.1438929