Question

Como calcular o valor da equação log(x-2)na base 3 - log(x-2) na base 9=1

Answer 1

Boa tarde Robinnho 

y = x - 2

log3(y) - log9(y) = 1

log(y)/log(3) - log(y)/2log(3) = 1

2log(y)/2log(3) - log(y)/2log(3) = 1

log(y) = 2log(3) 
log(y) = log(3^2) = log(9) 

y = 9 

y = x - 2 = 9

x = 9 + 2 = 11
.

Answer 2

A seguinte propriedade será útil no logaritmo de base 9

$\fbox{ \ell og_{a^C}B=\frac{1}{C}\cdot\ell og_aB~~~~(0\ \textless \ a\neq1) }$

Observe

$\ell og_{3}(x-2)-\ell og_{9}(x-2)=1\\\\\ell og_3(x-2)-\ell og_{3^2}(x-2)=1\\\\\ell og_3(x-2)-\frac{1}{2}\ell og_{3}(x-2)=1\\\\(\ell og_3(x-2))\cdot(1-\frac{1}{2})=1\\\\(\ell og_3(x-2))\cdot(\frac{1}{2})=1\\\\\ell og_3(x-2)=2$

Usando a definição de logaritmos

$\fbox{ y=\ell og_aB~\Longleftrightarrow~a^y=B~~~~(0\ \textless \ a\neq1) }$

Portanto

$\ell og_3(x-2)=2~\Longleftrightarrow~3^2=x-2\\\\9=x-2\\\\x=11~~~~(\textsf{resposta})$