Question

O resultado da equação exponencial 2^x+3 + 4^x-1 =17 tem solução para?

Answer 1

Olá,

use as propriedades da exponenciação:

$2^{x+3}+4^{x-1}=17\\ 2^x\cdot2^3+4^x\cdot4^{-1}=17\\\\ 8\cdot2^x+(2^2)^x\cdot \dfrac{1}{4} -17=0\\\\ \dfrac{1}{4}\cdot (2^x)^2+8\cdot2^x-17=0\\\\ 2^x=y\\\\ \dfrac{1}{4} y^2+8y-17=0\\\\ \Delta=8^2-4\cdot\left( \dfrac{1}{4} \right)\cdot(-17)\\\\ \Delta=64+17\\ \Delta=81$

$y= \dfrac{-8\pm \sqrt{81} }{2\cdot \dfrac{1}{4} }= \dfrac{-8\pm9}{ \dfrac{1}{2} }\begin{cases}y_1= \dfrac{-8+9}{ \dfrac{1}{2} } = \dfrac{1}{ \dfrac{1}{2} }= 2\\\\ y_2 =\dfrac{-8-9}{ \dfrac{1}{2} }= \dfrac{-17}{ \dfrac{1}{2} }=-34\notin\mathbb{R}\end{cases}\\\\\\ 2^x=y\\\\\\ 2^x=2\\ \not2^x=\not2^1\\\\ x=1\\\\\\ \huge\boxed{\text{S}=\{1\}}$

Tenha ótimos estudos ;P