Question

Questão do ITA
Dê o conjunto verdade da equação exponencial
3.5^x^2+3^x^2+1-8.3^x^2=0

Answer 1

Olá Luis.



Organizando e resolvendo a equação.

$\mathsf{3\cdot5^{x^2}+3^{x^2+1}-8\cdot3^{x^2}=0}\\\\\mathsf{3\cdot5^{x^2}+3^{x^2+1}-(5+3)\cdot3^{x^2}=0}\\\\\mathsf{3\cdot5^{x^2}+3^{x^2+1}-(5\cdot3^{x^2}+3\cdot3^{x^2})=0}\\\\\mathsf{3\cdot5^{x^2}+\diagup\!\!\!\!3^{x^2+1}-5\cdot3^{x^2}-\diagup\!\!\!\!3^{x^2+1}=0}\\\\\mathsf{3\cdot5^{x^2}-5\cdot3^{x^2}=0}\\\\\mathsf{3\cdot5^{x^2}=5\cdot3^{x^2}}\\\\\mathsf{\ell og(3\cdot5^{x^2})=\ell og(5\cdot3^{x^2})}$

$\mathsf{\ell og(3)+\ell og(5^{x^2})=\ell og(5)+\ell og(3^{x^2})}\\\\\mathsf{\ell og(3)+x^2\cdot\ell og(5)=\ell og(5)+x^2\cdot\ell og(3)}\\\\\mathsf{\ell og(3)-\ell og(5)=x^2\cdot\ell og(3)-x^2\ell og(5)}\\\\\mathsf{\ell og(3)-\ell og(5)=x^2\cdot(\ell og(3)-\ell og(5))}\\\\\mathsf{-1\cdot(\ell og(5)-\ell og(3))+x^2\cdot(\ell og(5)-\ell og(3))=0}\\\\\mathsf{(x^2-1)\cdot(\ell og(5)-\ell og(3))=0}$

$\mathsf{(x+1)\cdot(x-1)\cdot(\ell og(5)-\ell og(3))=0}\\\\\boxed{\mathsf{x=\pm1}}$


Dúvidas? comente.