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Hilbert’s problems form a list of twenty-three problems in mathematics published by David Hilbert, a German mathematician, in 1900. The problems were all unsolved at the time and several of them were very influential for 20th century mathematics. Hilbert believed it was essential for mathematicians to find new machineries and methods in order to solve the mentioned problems. The seventh problem deals with the transcendence of \(\alpha ^\beta \) for algebraic \(\alpha \ne 0,1\) and irrational algebraic \(\beta \). This problem was solved by Gelfond and (independently) Schneider. In 1935, Gelfond found a lower bound for the absolute value of the linear form $$\varLambda =\beta _1\log \alpha _1+\beta _2\log \alpha _2\ne 0.$$ He proved that $$\log |\varLambda |\gg -h(\varLambda )^\kappa ,$$ where \(h(\varLambda )\) is logarithmic height of the linear form \(\varLambda \), \(\kappa >5\) and \(\gg \) denotes inequality that is valid up to an unspecified constant factor. He noticed that generalization of his results could prove a huge amount of unsolved problems in number theory.