Question Number 20245 by virus last updated on 24/Aug/17
$$\mathrm{y}=\mathrm{2}^{\frac{\mathrm{1}}{{log}_{{x}} \mathrm{8}}} \\ $$$${then}\:{x}=? \\ $$
Answered by Tinkutara last updated on 24/Aug/17
$${y}\:=\:\mathrm{2}^{\mathrm{log}_{\mathrm{8}} \:{x}} \:=\:\mathrm{2}^{\frac{\mathrm{1}}{\mathrm{3}}\:\mathrm{log}_{\mathrm{2}} \:{x}} \\ $$$${y}^{\mathrm{3}} \:=\:\mathrm{2}^{\mathrm{log}_{\mathrm{2}} \:{x}} \:=\:{x} \\ $$$$\Rightarrow\:{x}\:=\:{y}^{\mathrm{3}} \\ $$
Answered by Rauny last updated on 27/Jan/19
$$\frac{\mathrm{1}}{\left(\frac{{ln}\:\mathrm{8}}{{ln}\:{x}}\right)}={log}_{\mathrm{2}} \:{y} \\ $$$$\frac{{ln}\:{x}}{{ln}\:\mathrm{8}}=\frac{{ln}\:{y}}{{ln}\:\mathrm{2}} \\ $$$${ln}\:{x}=\frac{{ln}\:\mathrm{8}}{{ln}\:\mathrm{2}}\centerdot{ln}\:{y}={log}_{\mathrm{2}} \:\mathrm{8}\centerdot{ln}\:{y} \\ $$$$=\mathrm{3}{ln}\:{y} \\ $$$${x}={e}^{\mathrm{3}{ln}\:{y}} \\ $$$$={y}^{\mathrm{3}} \\ $$$$\therefore{x}={y}^{\mathrm{3}} \\ $$