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Question Number 10912 by geovane10math last updated on 01/Mar/17

5^(log_2 3) is transcendental? General: Let a,b and c algebraic and log_b c transcendental. If a^(log_b c) is algebraic, so b = a^q , with q rational?

$$\mathrm{5}^{\mathrm{log}_{\mathrm{2}} \mathrm{3}} \:\mathrm{is}\:\mathrm{transcendental}? \\ $$$$\mathrm{General}: \\ $$$$\mathrm{Let}\:{a},{b}\:\mathrm{and}\:{c}\:\mathrm{algebraic}\:\mathrm{and}\:\mathrm{log}_{{b}} {c}\: \\ $$$$\mathrm{transcendental}.\:\mathrm{If}\:{a}^{\mathrm{log}_{{b}} {c}} \:\mathrm{is}\:\mathrm{algebraic},\:\mathrm{so} \\ $$$${b}\:=\:{a}^{{q}} ,\:\mathrm{with}\:{q}\:\mathrm{rational}? \\ $$

Commented by FilupS last updated on 03/Mar/17

A transendental number is one that is non alegebraic x=5^(log_2 3) one method of log_b (c) is trancendental if: a≠b (and a is prime)

$$\mathrm{A}\:\mathrm{transendental}\:\mathrm{number}\:\mathrm{is}\:\mathrm{one}\:\mathrm{that} \\ $$$$\mathrm{is}\:\mathrm{non}\:\mathrm{alegebraic} \\ $$$$\: \\ $$$${x}=\mathrm{5}^{\mathrm{log}_{\mathrm{2}} \mathrm{3}} \\ $$$$\mathrm{one}\:\mathrm{method}\:\mathrm{of}\:\mathrm{log}_{{b}} \left({c}\right)\:\mathrm{is}\:\mathrm{trancendental}\:\mathrm{if}: \\ $$$${a}\neq{b}\:\left(\mathrm{and}\:{a}\:\mathrm{is}\:\mathrm{prime}\right) \\ $$

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