Question

Se x=1+2^p e y=1+2^-p. Então y é igual a:

Answer 1

Aplicando logaritmo nas equações dadas:
$\begin{matrix}x=1+2^p\\x-1=2^p\\\log(x-1)=\log(2^p)\\\log(x-1)=p.\log(2)\end{matrix}\qquad\qquad\qquad\qquad\begin{matrix}y=1+2^{-p}\\y-1=2^{-p}\\\log(y-1)=\log(2^{-p})\\\log(y-1)=-p.\log(2)\\-\log(y-1)=p.\log(2)\end{matrix}$

Dessa forma, verificamos que:
$\log(x-1)=-\log(y-1)$

Desenvolvendo:
$\log(x-1)=-\log(y-1)\\\\\log(x-1)=\log\!\left[(y-1)^{-1}\right]\\\\\therefore\ {x-1}=(y-1)^{-1}\\\\\dfrac{1}{x-1}=y-1\\\\y=\dfrac{1}{x-1}+1\\\\\boxed{\boxed{y=\dfrac{x}{x-1}}}$