Question

Dada a função f(x)=log1/2x, então calcule o valor de f(1/8)

Answer 1

$\begin{array}{l}\mathsf{Sobre~a~quest\~ao:}\\\\\mathsf{Calcular~f(x)~quando~x=\frac{1}{8},~sendo~f(x)~uma~fun\c{c}\~ao~logar\'itimica~que}\\\mathsf{associa ~a~cada~x~o~n\'umero~\ell og_{\frac{1}{2}}(x).}\\\\\textsf{Substituindo x por 1/8 e aplicando as propriedades logar\'itimicas:}\end{array}$

$\begin{array}{l}\mathsf{\ell og_{\frac{1}{2}}(\frac{1}{8})~\Leftrightarrow~\ell og_{ \frac{1}{2}}(1)-\ell og_{\frac{1}{2}}(8)~\Leftrightarrow~0-\ell og_{ \frac{1}{2} }(8)~\Leftrightarrow~0-\ell og_{2^{-1}}(8)}\\\\\mathsf{0-\ell og_{2^{-1}}(2)^3~\Leftrightarrow~0-[3\ell og_{2^{-1}}(2)]}\end{array}$

$\begin{array}{l}\mathsf{Lembrando~do~seguinte~caso:~\large\fbox{ \mathsf{\ell og_{a^{\beta}}b~\Leftrightarrow~\frac{1}{\beta}\ell og_ab} }}\end{array}$

$\begin{array}{l}\mathsf{O~que~temos~\'e~o~seguinte:}\\\\\mathsf{0-[3\ell og_{2^{-1}}(2)]~\Leftrightarrow~0-[-\frac{1}{1}\cdot3\ell og_2(2)]~\Leftrightarrow~0-[-3\ell og_2(2)]}\\\\\mathsf{0-[-3\cdot1]~\Leftrightarrow~0-[-3]~\Leftrightarrow~0+3~\Leftrightarrow~\fbox{ 3 }}\end{array}$


$\begin{array}{l}\mathsf{Portanto,~nessa~fun\c{c}\~ao,~quando~x=\dfrac{1}{8},~~~f(x)=3.}\end{array}$