Question

Determine o conjunto da solução da equação: log ( 10x^2 + 5 ) - 2 log ( 1 - x ) = 1.

Answer 1

$\ell og(10x^2+5)-2\ell og(1-x)=1\\\\\ell og(10x^2+5)-\ell og(1-x)^2=1\\\\\ell og\begin{pmatrix}\dfrac{10x^2+5}{(1-x)^2}\end{pmatrix}=1\\\\\\\ell og\begin{pmatrix}\dfrac{10x^2+5}{1-2x+x^2}\end{pmatrix}=1~\Leftrightarrow~\dfrac{10x^2+5}{1-2x+x^2}=10^1\\\\\\10x^2+5=10\cdot(1-2x+x^2)\\\\10x^2\hspace{-15}\diagup+5=10-20x+10x^2\hspace{-15}\diagup\\\\-20x+10=5\\\\-20x=-5\\\\x=\dfrac{-5}{-20}\\\\\\\fbox{ x=\dfrac{1}{4} }\\\\\\\\S=\begin{Bmatrix}\dfrac{1}{4}\end{Bmatrix}~~\leftarrow~~conjunto~solu\c{c}\~ao$