Question

Considerando log 8 =A calcule, em função do A, o valor do log 20

Answer 1

$\displaystyle \sf \log 8=A \\\\ \log2^3 = A \\\\ 3\log 2=A \\\\ \log 2 = \frac{A}{3}$

Queremos log 20 em função de A. Façamos :

$\displaystyle \sf \log 20 = \log (4\cdot 5) \\\\ log(4\cdot 5) = \log 4+\log 5 \\\\ \log 4+\log 5 = \log 2^2 +\log \left(\frac{10}{2}\right) \\\\\\\log 2^2 +\log \left(\frac{10}{2}\right) = 2\log 2 +\log 10 -\log 2 \\\\\\ 2\log 2 +\log 10 -\log 2 = \log 2+\log 10 \\\\ \log 2+\log 10 = \frac{A}{3}+1\to \frac{A+3}{3} \\\\\\ Portanto : \\\\ \huge\boxed{\sf \log 20 = \frac{A+3}{3} }\checkmark$