Question

Me ajudem nessa equação exponencial: 5x + 125.5 -x = 30

Answer 1

Olá,

acompanhe a resolução passo a passo, aplicando as propriedades da exponenciação:

$\mathsf{5^x+125\cdot5^{-x}=30}\\\\ \mathsf{5^x+125\cdot \dfrac{1}{5^x} =30}\\\\ \mathsf{5^x+ \dfrac{125}{5^x}=30~~(Multiplique~os~extremos~por~5^x) }\\\\ \mathsf{(5^x)\cdot(5^x)+125=30\cdot(5^x)}\\\\ \mathsf{(5^x)^2-30\cdot5^x+125=0}\\\\ \mathsf{Fazemos~5^x=y}\\\\ \mathsf{y^2-30y+125=0~~(Eq.~do~2^o~grau)}\\\\ \mathsf{y_1=5~~e~~y_2=25}$

$\mathsf{5^x=y~Lembra?}\\\\ ~~\mathsf{5^x=5}~~~~~~~~~~~~~~~~\mathsf{5^x=25}\\ ~~\mathsf{5^x=5^1}~~~~~~~~~~~~~~~\mathsf{5^x=5^2}\\ \mathsf{\not5^x=\not5^1}~~~~~~~~~~~~~\mathsf{~\not5^x=\not5^2}\\\\ \mathsf{~~x_1=1}~~~~~~~~~~~~~~~~~\mathsf{x_2=2}\\\\\\ \Large\boxed{\mathsf{S=\{1,2\}}}$

Tenha ótimos estudos ;P