Question Number 223569 by Atomusmaths last updated on 29/Jul/25
Answered by Raphael254 last updated on 30/Jul/25
$$ \\ $$$$\mathrm{5}^{\mathrm{50}} \:=\:\left(\mathrm{5}^{{x}} \right)^{\frac{\mathrm{50}}{{x}}} \\ $$$$\mathrm{5}^{{x}} \:=\:\frac{\mathrm{50}}{{x}} \\ $$$${log}_{\mathrm{5}} \:\frac{\mathrm{50}}{{x}}\:=\:{x} \\ $$$${log}_{\mathrm{5}} \:\mathrm{50}\:−\:{log}_{\mathrm{5}} \:{x}\:=\:{x} \\ $$$${log}_{\mathrm{5}} \:\mathrm{2}×\mathrm{5}^{\mathrm{2}} \:−\:{log}_{\mathrm{5}} \:{x}\:=\:{x} \\ $$$${log}_{\mathrm{5}} \:\mathrm{2}\:+\:{log}_{\mathrm{5}} \:\mathrm{5}^{\mathrm{2}} \:−\:{log}_{\mathrm{5}} \:{x}=\:{x} \\ $$$${log}_{\mathrm{5}} \:\mathrm{2}\:+\:\mathrm{2}\:−\:{log}_{\mathrm{5}} \:{x}\:=\:{x} \\ $$$${log}_{\mathrm{5}} \:\mathrm{2}\:+\:\mathrm{2}\:=\:{log}_{\mathrm{5}} \:{x}\:+\:{x} \\ $$$$ \\ $$$${x}\:=\:\mathrm{2} \\ $$$$ \\ $$$$\left(\mathrm{5}^{{x}} \right)^{\frac{\mathrm{50}}{{x}}} \:=\left(\mathrm{5}^{\mathrm{2}} \right)^{\frac{\mathrm{50}}{\mathrm{2}}} =\:\left(\mathrm{25}\right)^{\mathrm{25}} \:=\:{t}^{{t}} \:\Rightarrow\:{t}\:=\:\mathrm{25} \\ $$
Answered by mr W last updated on 30/Jul/25
$${t}^{{t}} =\mathrm{5}^{\mathrm{50}} =\mathrm{5}^{\mathrm{2}×\mathrm{25}} =\left(\mathrm{5}^{\mathrm{2}} \right)^{\mathrm{25}} =\mathrm{25}^{\mathrm{25}} \\ $$$$\Rightarrow{t}=\mathrm{25} \\ $$
Answered by Abdulazim last updated on 19/Sep/25
$${t}^{{t}} =\mathrm{5}^{\mathrm{50}} \\ $$$${t}^{{t}} =\mathrm{5}^{\mathrm{2}\centerdot\mathrm{25}} \:\rightarrow\:{a}^{{n}\centerdot{m}} =\left({a}^{{n}} \right)^{{m}} \\ $$$${t}^{{t}} =\left(\mathrm{5}^{\mathrm{2}} \right)^{\mathrm{25}} \\ $$$${t}^{{t}} =\mathrm{25}^{\mathrm{25}} \\ $$$${t}=\mathrm{25} \\ $$