Question

Resolva: log3(x+1) - log9(x+1)=1 ?

Answer 1

$log_{_3}(x+1)-log_{_9}(x+1)=1\\\\Mundando\;a\;base\;9\;com\;a\;propriedade:\;log_{b}a=\frac{log_ca}{log_cb}\\\\log_{_3}(x+1)-\frac{log_{_3}(x+1)}{log_{_3}9}=1\\\\log_{_3}(x+1)-\frac{log_{_3}(x+1)}{2}=1\\\\log_{_3}(x+1)-\frac{1}{2}log_{_3}(x+1)=1\\\\Utilizando\;a\;propriedade:\;log\,a^c=c.log\,a\\\\log_{_3}(x+1)-log_{_3}(x+1)^{\frac{1}{2}}=1\\\\Utilizando\;a\;propriedade:\;log\frac{a}{c}=log\,a-log\,c\\\\log_{_3}\frac{x+1}{(x+1)^\frac{1}{2}}=1$

$\frac{x+1}{(x+1)^\frac{1}{2}}=3^1\\\\(x+1).(x+1)^{-\frac{1}{2}}=3\\\\(x+1)^{1-\frac{1}{2}}=3\\\\(x+1)^{\frac{1}{2}}=3\\\\x+1=3^2\\\\x+1=9\\\\x=8$