Question

resolva a equação log2 (x2+2x) = 3

Answer 1

Aplicando a definição de logaritmo

$\large\fbox{ a^y=b~\Leftrightarrow~y=\ell og_a(b) }~~sendo~~0\ \textless \ a \neq 1~~~e~~~0\ \textless \ b$

O que temos é:

$log_2(x^2+2x)=3\\\\2^3=x^2+2x\\\\8=x^2+2x\\\\0=x^2+2x-8~\Leftrightarrow~\underbrace{x^2+2x-8=0}_{equa\c{c}\~ao~quadr\'atica}$

Resolvendo a equação quadrática por Bhaskara

$\dfrac{-b\pm\sqrt{\Delta}}{2a}~~~onde~~\Delta=b^2-4ac$

Calculando primeiro o discriminante

$\Delta=2^2-4\cdot1\cdot(-8)\\\\\Delta=4-(-32)\\\\\Delta=36$

Substituindo Δ na fórmula

$\dfrac{-(2)\pm\sqrt{36}}{~~2\cdot1}\\\\\\~~~~\dfrac{-2\pm6}{~~2}\\\\\\\\\left\{\begin{matrix} \hspace{-15}{x_1=\dfrac{-2+6}{~~2}=\dfrac{4}{2}=2}\\\\\\x_2=\dfrac{-2-6}{~~2}=\dfrac{-8}{~~2}=-4\end{matrix}\right.$


Solução


$x=\left\{\begin{matrix} -4\\\\ou\\\\\hspace{-3}{2} \end{matrix}\right.$