leimkuhl@uiucdcsp.UUCP (02/21/85)
The problem was to solve y = x*z^x for x in terms of y and z. Solution: x * z^x = x * z * z^(x-1) = z * x * z^(x-1) = z * d(z^x)/dz Thus we would like a solution to the problem y/z = d(z^x)/dz Int( s=1,z; yds/s ) = z^x - 1^x = z^x - 1 or, ylog(z) = z^x - 1 Solving for x, we have x = log( y*log(z) + 1 ) / log(z) | | | | | | | | | | | | | | | | | | | | | | | Did you find the flaw in that derivation? Be honest! It is true that the above solution for x solves y = z * d(z^x)/dz, but that does not make it a solution of y = x * z^x. The reason is that x is a function of z (and y, too, but we can assume y is constant). Thus the first step in the above derivation is wrong: y = x * z^x d(z^x)/dz =>> y = z * --------- (here d/dz denotes partial wrt z) dx/dz If we rewrite, letting z = t, y = a, x=f(t), we get a more easily understandable problem (we are so conditioned to regard x as a variable!) f(t) * t^f(t) = a (*) Which yields, with the above technique, af'/t = d(t^f(t))/dt Which is not so easily dealt with. If we take logarithms of both sides of (*), divide through by f, and collect terms, we get log(f/a)/f = -log(t) But this is the only thing that occurs. Almost certainly you will have to solve such a nonlinear implicit equation as (*) by numerical approximation (or series). I haven't thought about what a good numerical method might be for this problem. -Ben Leimkuhler