 Gelfond–Schneider theorem

For algebraic $a\ne 0,1$ and algebraic irrational $b$, every value $\exp (b\lambda )$ with $e^{\lambda }=a$ is transcendental. Consequently any quotient of chosen nonzero logarithms of algebraic numbers is either rational or transcendental.

The Gelfond–Schneider theorem is a fundamental result in transcendental number theory, established by Aleksandr Gelfond and Richard Schneider in the 1930s.