Question

Resolva as equações exponenciais abaixo:

Answer 1

Explicação passo-a-passo:

a) $\sf 2^{3x-1}=32$

$\sf 2^{3x-1}=2^5$

$\sf 3x-1=5$

$\sf 3x=5+1$

$\sf 3x=6$

$\sf x=\dfrac{6}{3}$

$\sf x=2$

$\sf S=\{2\}$

b) $\sf 11^{2x+5}=1$

$\sf 11^{2x+5}=11^0$

$\sf 2x+5=0$

$\sf 2x=-5$

$\sf x=\dfrac{-5}{2}$

$\sf S=\left\{\dfrac{-5}{2}\right\}$

c) $\sf 3^{x^2+2x}=243$

$\sf 3^{x^2+2x}=3^5$

$\sf x^2+2x=5$

$\sf x^2+2x-5=0$

$\sf \Delta=2^2-4\cdot1\cdot(-5)$

$\sf \Delta=4+20$

$\sf \Delta=24$

$\sf x=\dfrac{-2\pm\sqrt{24}}{2\cdot1}=\dfrac{-2\pm2\sqrt{6}}{2}$

$\sf x'=\dfrac{-2+2\sqrt{6}}{2}~\longrightarrow~x'=-1+\sqrt{6}$

$\sf x"=\dfrac{-2-2\sqrt{6}}{2}~\longrightarrow~x"=-1-\sqrt{6}$

$\sf S=\{-1+\sqrt{6},-1-\sqrt{6}\}$