Question

equação exponencial
a) $7^{x}+7^{x-1} = 8$
b) $5^{x} -125. 5^{-x} =30$

Answer 1

$7^{x}+7^{x-1} = 8\\\\7^{x}+7^{x}*7^{-1} = 8\\\\ 7^x*(1+7^{-1})=8\\\\7^x*(1+\frac{1}{7})=8\\\\7^x* \frac{8}{7} =8 \\\\ \frac{7^x}{7}= \frac{8}{8} \\\\7^{x-1}=1\\\\x-1=0 \\\\\boxed{x=1}$


b)
$5^{x} +125* 5^{-x} =30\\\\ 5^x+ \frac{125}{5^x}=30 \\\\ \frac{5^{2x}+125}{5^x}=30\\\\ 5^{2x}+125=30*5^x\\\\ \boxed{\boxed{5^{2x}-30*5^x +125=0}}$

transformando em uma equação do segundo grau
u=5^x

$u^2-30u+125=0 \\\\ u= \frac{-(-30)\pm \sqrt{(-30)^2-4*1*(125)} }{2*1} \\\\u= \frac{30\pm20}{2} \\\\u_1= \frac{30-20}{2}=5 \\\\u_2= \frac{30+20}{2}= 25$

resolvendo
$u=5\\5^x=5 \to \boxed{x=1}\\\\\\u=25\\5^x=25\\5^x=5^2\to \boxed{x=2}$