Question Number 5185 by Yozzii last updated on 28/Apr/16
$${If}\:\:\mathrm{2}^{{x}\:} \:{and}\:\:\mathrm{3}^{{x}} \:{are}\:{integers}\:{for}\:{some} \\ $$$${x}\in\mathbb{R}^{+} ,\:{must}\:{x}\:{be}\:{an}\:{integer}?\: \\ $$
Answered by 123456 last updated on 28/Apr/16
$$\mathrm{2}^{{x}} ={a}\Leftrightarrow{x}=\mathrm{log}_{\mathrm{2}} {a} \\ $$$$\mathrm{3}^{{x}} ={b}\Leftrightarrow{x}=\mathrm{log}_{\mathrm{3}} {b} \\ $$$$\mathrm{log}_{\mathrm{2}} {a}=\mathrm{log}_{\mathrm{3}} {b} \\ $$$${k}\in\mathbb{Z} \\ $$$${a}=\mathrm{2}^{\mathrm{log}_{\mathrm{3}} {b}} ,{a}\in\mathbb{Z}\Rightarrow\mathrm{log}_{\mathrm{3}} {b}={k}\Leftrightarrow{b}=\mathrm{3}^{{k}} \Rightarrow{x}\in\mathbb{Z} \\ $$$${b}=\mathrm{3}^{\mathrm{log}_{\mathrm{2}} {a}} ,{b}\in\mathbb{Z}\Rightarrow\mathrm{log}_{\mathrm{2}} {a}={k}\Leftrightarrow{a}=\mathrm{2}^{{k}} \Rightarrow{x}\in\mathbb{Z} \\ $$
Commented by Yozzii last updated on 28/Apr/16
$${In}\:{line}\:\mathrm{5},\:{a}\in\mathbb{Z}\:{I}\:{think}\:{could}\:{also} \\ $$$${mean}\:\mathrm{2}^{{log}_{\mathrm{3}} {b}} =\mathrm{2}^{{log}_{\mathrm{2}} {n}} ={n}\:{for}\:{some}\:{n}\in\mathbb{Z}. \\ $$$$\Rightarrow{b}=\mathrm{3}^{{log}_{\mathrm{2}} {n}} \: \\ $$$${But}\:{b},{n}\in\mathbb{Z}^{+} \Rightarrow{n}=\mathrm{2}^{{j}} \in\mathbb{Z}\Rightarrow{x}={j}\in\mathbb{Z} \\ $$$$ \\ $$$$\mathrm{3}^{{log}_{\mathrm{2}} {a}} =\mathrm{3}^{{log}_{\mathrm{3}} {q}} \Rightarrow{a}=\mathrm{2}^{{log}_{\mathrm{3}} {q}} \\ $$$$\Rightarrow{q}=\mathrm{3}^{{r}} \Rightarrow{x}={r}\in\mathbb{Z}^{+} . \\ $$