Question

função exponenciais :

2.3^x + 5.3^x-1 =4.3^x+1 -75

Answer 1

Falando que 3^x = y, temos que:

2.3^x + 5.3^(x-1) = 4.3^(x+1)-75

2.3^x + 5.3^x.3^(-1) = 4.3^x.3^1 - 75

2y + 5y/3 = 12y - 75

2y+5y/3-12y = -75

(6y+5y-36y)/3 = -75

-25y = -75*3

y = -75*3 / -25

y = 3*3

y = 9

Voltando:

3^x = y

3^x = 9
3^x = 3^2

x = 2

Answer 2

Ae Lucas,

as mesmas propriedades da exponenciação já utilizadas:

$\mathsf{1^a\Rightarrow a^{m+n}=a^m\cdot a^n}\\\\ \mathsf{2^a \Rightarrow a^{-m} =\dfrac{1}{a^m} }$

..............................

$\mathsf{2\cdot3^x+5\cdot3^{x-1}=4\cdot3^{x+1}-75}\\ \mathsf{2\cdot3^x+5\cdot3^x\cdot3^{-1}=4\cdot3^x\cdot3^1-75}\\\\ \mathsf{2\cdot3^x+5\cdot3^x\cdot \dfrac{1}{3^1}=4\cdot3^x\cdot3-75 }\\\\ \mathsf{2\cdot3^x+5\cdot \dfrac{1}{3}\cdot3^x=4\cdot3\cdot3^x-75 }\\\\ \mathsf{2\cdot3^x+ \dfrac{5}{3}\cdot3^x=12\cdot3^x-75 }\\\\ \mathsf{2\cdot3^x+ \dfrac{5}{3}\cdot3^x-12\cdot3^x=-75 }\\\\ \mathsf{3^x~em~evidencia:}$

$\mathsf{3^x\cdot\left(2+ \dfrac{5}{3}-12\right)=-75}\\\\ \mathsf{3^x\cdot\left(- \dfrac{25}{3}\right)=-75 }\\\\ \mathsf{3^x=-75\div\left( -\dfrac{25}{3}\right)}\\\\ \mathsf{3^x=-75\cdot\left(- \dfrac{3}{25}\right) }\\\\ \mathsf{3^x= \dfrac{225}{25} }\\\\ \mathsf{3^x=9}\\ \mathsf{3^x=3^2}\\ \mathsf{\not3^x=\not3^2}\\\\ \mathsf{x=2}\\\\\\ \huge\boxed{\mathsf{S=\{2\}}}$

Tenha ótimos estudos mano ;D