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SolveTwoPowersOfNumericBases(AngouriMath.Entity,AngouriMath.Entity.Variable)

Method (no overloads)

Summary

a ^ p(x) = b ^ q(x) for numeric a and b, solved by taking
logarithms, which keeps the answer exact.

Remarks

SolveMultiplicative(AngouriMath.Entity,AngouriMath.Entity.Variable) reaches these by substitution: it divides one
exponent by the other and simplifies, and for two different integer bases that ratio
is ln(3)/ln(2) — irrational — so InnerSimplified settles it to a decimal
and everything downstream is numeric. The answer then agrees with the exact one to
about seventeen figures and diverges after, which is a double promoted to a
decimal rather than a number that was computed.
#1007 Taking logarithms has no such step: a ^ p = b ^ q is p ln a = q ln b for positive real a and b, and that is an ordinary equation the
analytical solver answers exactly — 3 ^ (x+1) = 2 ^ (x-1) becomes
-ln(6) / ln(3/2).
Both bases must be decidably positive reals, which is what makes ln of
them real and the step an equivalence rather than a branch choice. Anything else
declines and the substitution path still gets its turn, so this only ever adds
answers.

AngouriMath

Table of contents

AngouriMath

SolveTwoPowersOfNumericBases​(AngouriMath.​Entity,​AngouriMath.​Entity.​Variable)

SolveTwoPowersOfNumericBases(AngouriMath.Entity,AngouriMath.Entity.Variable)