Re: [Maxima-discuss] Why can't Maxima solve 5^(3*x+2) = 7^(3*x+2) symbolically? PEBCAK or a limitation?

[Matthias A. S. <mat...@we...>]

Picking up Eduardo’s idea (https://sourceforge.net/p/maxima/mailman/message/59300108/), I wrote, admittedly with the help of AI, a symbolic wrapper around the logarithmic lifting

	a^{u(x)} = b^{v(x)}
	u(x)\log a - v(x)\log b = 2\pi i k

which Maxima fails to expose automatically. I’ve attached the file below, and here is what the program does on two exponential equations.
 
(%i1)  kill(all)$
(%i1)  load("all_exp_eq_solutions_v1.0.0.mac")$

(%i2)  eq1: 5^(3*x+2) = 7^(3*x+2)$
(%i3)  sols1: all_solutions(eq1, x)$
(%i4)  real(eq1, x);
(%o4) x=-(2/3)
(%i5)  complex_report(eq1, x);
(%o5) [x=-((2*i*π*k+2*log (7)-2*log (5))/(3*log (7)-3*log (5))),x=-((2*i*π*k)/(3*log (7/5)))-2/3]
(%i6)  complex(eq1, x);
(%o6) x=-((2*i*π*k)/(3*log (7/5)))-2/3

(%i7)  eq2: 5^(3*x+2) = 7^(2*x+1)$
(%i8)  sols2: all_solutions(eq2, x)$
(%i9)  real_solution(eq2, x);
(%o9) x=-((log (7)-2*log (5))/(2*log (7)-3*log (5)))
(%i10)  complex_report(eq2, x);
(%o10) x=-((2*i*π*k+log (7)-2*log (5))/(2*log (7)-3*log (5)))

I’m sure there’s room for improvement on the code level.  As Maxima quite often returns equations unsolved, where other CAS like Maple’s solve(eq1,x) or SolveTools:-ComplexSolve({eq1}) return solutions, this wrapper could serve as a starting point for a planned “Maxima equation toolkit” in the absence of a reduce() function in Maxima.

Cheers, Tilda