Question

Calcule o valor de x: 5^x -75.5^-x = 22

Answer 1

$Ola'! \\ \\ 5^{x}-75. 5^{-x}=22 \\ Resoluc\~ao: \\ \\ 5^{x}- \frac{75}{ 5^{x} }=22 \\ \\ 5^{2x}-75=22. 5^{x} \\ \\ 5^{2x}-22. 5^{x}-75=0~~~--\ \textgreater \ fazendo~~ \boxed{5^{x}=m} \\ \\ m^{2}-22m-75=0 \\ \\ (m-25)(m+3)=0~~--\ \textgreater \ igualamos~os~produtos \\ \\ (m-25)=0~~~ou~~~~(m+3)=0 \\ \\ \boxed{m=25~~~~ou~~~~m=-3}~~~--\ \textgreater \ como~tinhamos~chamados~m= 5^{x} \\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~voltamos, veja: \\ m=25=\ \textgreater \ 5^{x}=25 \\ ~~~~~~~~~~~=\ \textgreater \ 5^{x}=5\² \\ ~~~~~~~~~~~=\ \textgreater \ \boxed{x=2} \\ .$