Find-n-in-this-equation-2-n-4096-

Question Number 106630 by zahaku last updated on 06/Aug/20

$${Find}\:{n}\:{in}\:{this}\:{equation}: \\ $$$$\left(−\mathrm{2}\right)^{{n}} \:=\:\mathrm{4096} \\ $$

Answered by bemath last updated on 06/Aug/20

(−2)^n =(1024)×4=(2)^(12) =(−2)^(12) →n = 12 . @bemath@

$$\left(−\mathrm{2}\right)^{\mathrm{n}} =\left(\mathrm{1024}\right)×\mathrm{4}=\left(\mathrm{2}\right)^{\mathrm{12}} =\left(−\mathrm{2}\right)^{\mathrm{12}} \\ $$$$\rightarrow\mathrm{n}\:=\:\mathrm{12}\:.\:@\mathrm{bemath}@ \\ $$

Commented by zahaku last updated on 06/Aug/20

$${There}\:{isn}'{t}\:{another}\:{way}\:{to}\:{find}\:{n} \\ $$$${in}\:{the}\:{same}\:{equation}\:? \\ $$

Answered by Aziztisffola last updated on 06/Aug/20

n is even ⇒n=2p (−2)^n = 4096⇔ (−2)^(2p) =4096 ⇔4^p =4096 ⇒ln(4^p )=ln(4096) ln(4)p=ln(4096) ⇒p=((ln(4096))/(2ln(2))) 2p=((ln(4096))/(ln(2)))=12 2p=12=n ⇒n=12.

$$\:{n}\:{is}\:{even}\:\Rightarrow{n}=\mathrm{2}{p} \\ $$$$\:\left(−\mathrm{2}\right)^{{n}} \:=\:\mathrm{4096}\Leftrightarrow\:\left(−\mathrm{2}\right)^{\mathrm{2}{p}} =\mathrm{4096} \\ $$$$\:\Leftrightarrow\mathrm{4}^{\mathrm{p}} =\mathrm{4096}\:\Rightarrow\mathrm{ln}\left(\mathrm{4}^{\mathrm{p}} \right)=\mathrm{ln}\left(\mathrm{4096}\right) \\ $$$$\mathrm{ln}\left(\mathrm{4}\right)\mathrm{p}=\mathrm{ln}\left(\mathrm{4096}\right)\:\Rightarrow\mathrm{p}=\frac{\mathrm{ln}\left(\mathrm{4096}\right)}{\mathrm{2ln}\left(\mathrm{2}\right)} \\ $$$$\:\mathrm{2p}=\frac{\mathrm{ln}\left(\mathrm{4096}\right)}{\mathrm{ln}\left(\mathrm{2}\right)}=\mathrm{12} \\ $$$$\:\mathrm{2p}=\mathrm{12}=\mathrm{n}\:\Rightarrow\mathrm{n}=\mathrm{12}. \\ $$

Commented by zahaku last updated on 06/Aug/20