Question

Determine o valor da expressao log5 28 considerando log 2=0,3 log 5=0,7 e log log 7=0,8.?

Answer 1

$\mathsf{Informa\c{c}\~oes:}~\begin{Bmatrix}\mathsf{\ell og(2)\approx0,3}\\\mathsf{\ell og(5)\approx0,7}\\\mathsf{\ell og(7)\approx0,8}\end.\\\\\\\\\mathsf{Propriedades:}~\begin{Bmatrix}\mathsf{Do~produto:\ell og_{\alpha}(\beta\cdot\gamma)~\Leftrightarrow~\ell og_{\alpha}(\beta)+\ell og_{\alpha}(\gamma)}\\\\\mathsf{Da~pot\^encia:\ell og_{\alpha}(\beta)^{\gamma}~\Leftrightarrow~\gamma\cdot\ell og_{\alpha}(\beta)}\end.\\\\\\\textsf{Aplicando...}$

$\mathsf{\ell og_5(28)}\\\\\mathsf{\ell og_5(4\cdot7)}\\\\\mathsf{\ell og_5(4)+\ell og_5(7)}\\\\\mathsf{\ell og_5(2^2)+\ell og_5(7)}\\\\\mathsf{2\ell og_5(2)+\ell og_5(7)}\\\\\\\textsf{Lembrando que:}~\fbox{ \mathsf{\ell og_{a}(b)=\dfrac{\ell og(b)}{\ell og(a)}} }$

$\begin{array}{l}\mathsf{2\ell og_5(2)+\ell og_5(7)}\\\\\mathsf{\dfrac{2\ell og(2)}{\ell og(5)}+\dfrac{\ell og(7)}{\ell og(5)}}\\\\\mathsf{\approx\dfrac{2\cdot(0.3)}{0.7}+\dfrac{0.8}{0.7}}\\\\\mathsf{\approx\dfrac{0.6}{0.7}+\dfrac{0.8}{0.7}}\\\\\mathsf{\approx\dfrac{1.4}{0.7}}\\\\\fbox{ \mathsf{\approx2} }\end{array}$