Question

Resolva a equação exponencial

8 -³x-² = 32 -5x+²

Answer 1

Propriedades utilizadas

$\bullet\,\,\mathsf{(a^{m})^{n}=a^{m\cdot n}}\\\\\bullet\,\,\mathsf{se\,\,a\,\,\'e\,\,positivo\,\,e\,\,diferente\,\,de\,\,1,\,\,ent\~ao\,\,a^{m}=a^{n}\,\,implica\,\,em\,\,m=n}$
______________________________

$\mathsf{8^{-3x-2}=32^{-5x+2}}$

Vamos encontrar a fatoração de 8 e 32 em primos:

8 | 2
4 | 2
2 | 2
1

Logo 8 = 2 . 2 . 2 = 2³

32 | 2
16 | 2
08 | 2
04 | 2
02 | 2
1

Logo 32 = 2 . 2 . 2 . 2 . 2 = 2⁵

Com isso,

$\mathsf{8^{-3x-2}=32^{-5x+2}}\\\\\mathsf{(2^{3})^{-3x-2}=(2^{5})^{-5x+2}}\\\\\mathsf{2^{3\cdot(-3x-2)}=2^{5\cdot(-5x+2)}}\\\\\mathsf{2^{-9x-6}=2^{-25x+10}}$

As bases são iguais, positivas e diferentes de 1, então podemos igualar os expoentes:

$\mathsf{-9x-6=-25x+10}\\\\\mathsf{-9x+25x=10+6}\\\\\mathsf{16x=16}\\\\\boxed{\boxed{\mathsf{x=1}}}$