It's straightforward that $t$ must be irrational. I have googled many variations of this question and browsed through some books on transcendental number theory. There is much that is said about when the base is the same, but not for when the power is the same like here.

For example, does $-\sqrt 2$ satisfy this? I would be happy even with some buzz words about what machinery covers this question, or whether it is part of some grand and long-term conjecture in transcendental number theory now far beyond our powers.