Question

Na expressão log 25 (0,2) = z, z vale:
A) z=1/2
B) 2=-1/2
C) z=-5
D) z=-2
E) z=0

Answer 1

$\large\begin{array}{l}\ell og_{25}~(0.2)~\Leftrightarrow~\ell og_{25}~\dfrac{5}{25}~ ~\Leftrightarrow~\ell og_{5^2}~\dfrac{5}{25} ~\Leftrightarrow~\frac{1}{2}\ell og_5~\dfrac{5}{25}\\\\\\\frac{1}{2}\ell og_5~\dfrac{5}{25} ~\Leftrightarrow~\frac{1}{2}\begin{pmatrix}\ell og_5~5-\ell og_5~25\end{pmatrix}~\Leftrightarrow~\frac{1}{2}\begin{pmatrix}1-\ell og_5~5^2\end{pmatrix}\end{array}$


$\large\begin{array}{l}\frac{1}{2}\begin{pmatrix}1-2\ell og_5~5\end{pmatrix}~\Leftrightarrow~\frac{1}{2}\begin{pmatrix}1-2\cdot1 \end{pmatrix}~\Leftrightarrow~\frac{1}{2}(1-2)~\Leftrightarrow~\frac{1}{2}\cdot (-1)\\\\\\\frac{1}{2}\cdot(-1)=-\dfrac{1}{2}~~\checkmark\\\\\\\therefore~~\large\fbox{ \ell og_{25}(0.2)=-\dfrac{1}{2}=z }\end{array}$