Menu Close

Question-94868




Question Number 94868 by O Predador last updated on 21/May/20
Commented by john santu last updated on 21/May/20
n^2  = 2^(2n)  ⇒ n = 2^n    ln(n) = n ln(2)   ((ln(n))/n) = ln(2) ⇒ ln(n)^(1/n) =ln(2)  n^(1/n)  = 2 ; we can aply Lambert  W function
$$\mathrm{n}^{\mathrm{2}} \:=\:\mathrm{2}^{\mathrm{2n}} \:\Rightarrow\:\mathrm{n}\:=\:\mathrm{2}^{\mathrm{n}} \: \\ $$$$\mathrm{ln}\left(\mathrm{n}\right)\:=\:\mathrm{n}\:\mathrm{ln}\left(\mathrm{2}\right)\: \\ $$$$\frac{\mathrm{ln}\left(\mathrm{n}\right)}{\mathrm{n}}\:=\:\mathrm{ln}\left(\mathrm{2}\right)\:\Rightarrow\:\mathrm{ln}\left(\mathrm{n}\right)^{\frac{\mathrm{1}}{\mathrm{n}}} =\mathrm{ln}\left(\mathrm{2}\right) \\ $$$$\mathrm{n}^{\frac{\mathrm{1}}{\mathrm{n}}} \:=\:\mathrm{2}\:;\:\mathrm{we}\:\mathrm{can}\:\mathrm{aply}\:\mathrm{Lambert} \\ $$$$\mathrm{W}\:\mathrm{function}\: \\ $$
Commented by hknkrc46 last updated on 21/May/20
n^2 =4^n  ⇒ n^2 =(2^2 )^n  ⇒ n^2 =(2^n )^2  ⇒ n=2^n
$$\mathrm{n}^{\mathrm{2}} =\mathrm{4}^{\mathrm{n}} \:\Rightarrow\:\mathrm{n}^{\mathrm{2}} =\left(\mathrm{2}^{\mathrm{2}} \right)^{\mathrm{n}} \:\Rightarrow\:\mathrm{n}^{\mathrm{2}} =\left(\mathrm{2}^{\mathrm{n}} \right)^{\mathrm{2}} \:\Rightarrow\:\mathrm{n}=\mathrm{2}^{\mathrm{n}} \\ $$$$ \\ $$
Commented by O Predador last updated on 21/May/20
    Obrigado!
$$\: \\ $$$$\:\mathrm{Obrigado}! \\ $$
Commented by mr W last updated on 21/May/20
for n>0: 4^n =(2^n )^2 >n^2   therefore 2^n =4^n  has no solution!
$${for}\:{n}>\mathrm{0}:\:\mathrm{4}^{{n}} =\left(\mathrm{2}^{{n}} \right)^{\mathrm{2}} >{n}^{\mathrm{2}} \\ $$$${therefore}\:\mathrm{2}^{{n}} =\mathrm{4}^{{n}} \:{has}\:{no}\:{solution}! \\ $$
Commented by hknkrc46 last updated on 21/May/20
De nada !
$$\mathrm{De}\:\mathrm{nada}\:! \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *