Reduce[ForAll[{x, y, z}, {{x, y, z} > 0, 2^x == 3^y == 5^z, #}],
Reals] & /@ {2 x < 3 y < 5 z, 5 z < 2 x < 3 y, 3 y < 5 z < 2 x,
3 y < 2 x < 5 z}
Why does the result obtained using this method turn out to be incorrect?
{False, False, False, False}
The ForAll syntax to use in this case is ForAll[x,cond,expr], not ForAll[x,expr]:
Reduce[ForAll[{x, y, z},
{{x, y, z} > 0, 2^x == 3^y == 5^z},
3 y < 2 x < 5 z],
Reals]
True
The way you did it, you asked whether both expressions 2^x == 3^y == 5^z and 3 y < 2 x < 5 z are true for all positive triples, which is obviously false.
