Question

Se log 2 = 0,30 e log 3 = 0,48, determine os logaritmos abaixo utilizando propriedades operatórias:
a) log 72
b) log 125
c) log4 27

Answer 1

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$\large\boxed{\begin{array}{l}\tt a)~\sf \ell og72\\\begin{array}{c|l}\sf72&\sf2\\\sf36&\sf2\\\sf18&\sf2\\\sf9&\sf3\\\sf3&\sf3\\\sf1\end{array}\\\sf 72=2^3\cdot3^2\\\sf \ell og72=\ell og(2^3\cdot3^2)\\\sf \ell og72=\ell og2^3+\ell og3^2\\\sf \ell og72=3\ell og2+2\ell og3\\\sf \ell og72=3\cdot0,30+2\cdot0,48\\\sf \ell og72=0,90+0,96\\\sf \ell og72=1,86\end{array}}$

$\large\boxed{\begin{array}{l}\tt b)~\sf\ell og125\\\begin{array}{c|l}\sf125&\sf5\\\sf25&\sf5\\\sf5&\sf5\\\sf1\end{array}\\\sf \ell og125=\ell og5^3\\\sf \ell og125=3\ell og5\\\sf \ell og125= 3\ell og\bigg(\dfrac{10}{2}\bigg)\\\sf \ell og125=3[\ell og10-\ell og2]\\\sf \ell og125=3[1-0,30]\\\sf \ell og125=3\cdot0,7\\\sf \ell og125=2,1\end{array}}$

$\large\boxed{\begin{array}{l}\tt c)~\sf \ell og_427\\\begin{array}{c|l}\sf27&\sf3\\\sf9&\sf3\\\sf3&\sf3\\\sf1\end{array}\\\begin{array}{c|l}\sf4&\sf2\\\sf 2&\sf2\\\sf 1\end{array}\\\sf 27=3^3~~4=2^2\\\sf \ell og_427=\dfrac{\ell og27}{\ell og4}\\\sf \ell og_427=\dfrac{\ell og3^3}{\ell og2^2}\\\sf \ell og_427=\dfrac{3\ell og3}{2\ell og2}\\\sf\ell og_427=\dfrac{3\cdot0,48}{2\cdot0,30}\\\sf \ell og_427=\dfrac{1,44}{0,60}\\\sf \ell og_427=2,4 \end{array}}$