Question

Resolva a equação exponencial 5^x+125•5^-x=30​

Answer 1

Resposta:

Olá,

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

\begin{gathered}\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}\end{gathered}

5

x

+125⋅5

−x

=30

5

x

+125⋅

5

x

1

=30

5

x

+

5

x

125

=30 (Multiplique os extremos por 5

x

)

(5

x

)⋅(5

x

)+125=30⋅(5

x

)

(5

x

)

2

−30⋅5

x

+125=0

Fazemos 5

x

=y

y

2

−30y+125=0 (Eq. do 2

o

grau)

y

1

=5 e y

2

=25

\begin{gathered}\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}\\\\\\ < br / > \Large\boxed{\mathsf{S=\{1,2\}}}\end{gathered}

5

x

=y Lembra?

5

x

=5 5

x

=25

5

x

=5

1

5

x

=5

2



5

x

=



5

1



5

x

=



5

2

x

1

=1 x

2

=2

<br/>

S={1,2}

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Answer 2

Resposta:

$5^x+125\cdot 5^{-x}=30$

aplicar as propriedades dos expoentes:

$5^x+125\left(5^x\right)^{-1}=30$

reescrever a equação com 5ˣ= u

$u+125\left(u\right)^{-1}=30$

$u=25,\:u=5$

substitua u = 5ˣ  solucione para x

$x=2,\:x=1$

Resposta:  x=2, :x=1                                              

Espero ter ajudado  bons estudos....