Question

Resolva a seguinte equação: {log(b)a sendo b = base e a=logaritmando}

log(3)[x-2]-log(9)[x-2]=2

Answer 1

Vou desenvolver aplicando as principais propriedades dos logaritmos e então aplicar a definição:

$\ell og_{3}(x-2)-\ell og_{9}(x-2)=2\\\\\ell og_3(x-2)-\ell og_{3^2}(x-2)=2\\\\\ell og_3(x-2)-\frac{1}{2}\ell og_{3}(x-2)=2\\\\\ell og_3(x-2)-\ell og_3(x-2)^{\frac{1}{2}}=2\\\\\ell og_3(x-2)-\ell og_3\sqrt{(x-2)}=2\\\\\ell og_3(\frac{(x-2)}{\sqrt{(x-2)}})=2\\\\3^2=\frac{(x-2)}{\sqrt{(x-2)}}\\\\3^2=\frac{(x-2)\cdot\sqrt{(x-2)}}{(x-2)}\\\\3^2=\sqrt{(x-2)}\\\\3^4=x-2\\\\81=x-2\\\\x=83~~\leftarrow~\mathsf{resposta}$