Question

O conjunto solução da equação log2 ( x + 3 ) + log 2 ( x - 4 ) = 3 ?

Answer 1

$\ell og_2(x+3)+\ell og_2(x-4)=3\\\\\ell og_2(x+3)\cdot(x-4)=3\\\\\ell og_2(x^2-4x+3x-12)=3\\\\\ell og_2(x^2-x-12)=3\\\\x^2-x-12=2^3\\\\x^2-x-12=8\\\\\underbrace{x^2-x-20=0}_{equa\c{c}.~quadr\'atica}\\\\\\\Delta=b^2-4ac\\\\\Delta=(-1)^2-[4\cdot1\cdot(-20)]\\\\\Delta=1-[-80]\\\\\Delta=1+80\\\\\Delta=81$

$\dfrac{-b\pm\sqrt{\Delta}}{2a}\\\\\\\dfrac{-(-1)\pm\sqrt{81}}{2}\\\\\\\dfrac{1\pm9}{2}\\\\\\x_1=\dfrac{10}{2}=5\\\\\\x_2=-\dfrac{8}{2}=-4$

Atendendo as condições de existência do logaritmo, onde o logaritmando não pode ser negativo, a solução da equação será

$S=\begin{Bmatrix}5\end{Bmatrix}~~\leftarrow~~conjunto~solu\c{c}\~ao$