Question

Considerando as aproximações log2 = 0,3 e log 3 = 0,48, calcule: log da raiz cubica de 144

Answer 1

$\mathrm{\ell og\,}2=0,30;\;\;\mathrm{\ell og\,}3=0,48.$


Então, temos que

$\mathrm{\ell og\,}\sqrt[3]{144}\\ \\ =\mathrm{\ell og\,}(144^{1/3})\\ \\ =\dfrac{1}{3}\mathrm{\,\ell og\,}144$


Decompondo o número $144$ em seus fatores primos, temos

$=\dfrac{1}{3}\mathrm{\,\ell og\,}(2^{4}\cdot 3^{2})\\ \\ \\ =\dfrac{1}{3}\,[\mathrm{\ell og\,}(2^{4})+\mathrm{\ell og\,}(3^{2})]\\ \\ \\ =\dfrac{1}{3}\,[4\,\mathrm{\ell og\,}2+2\,\mathrm{\ell og\,}3]\\ \\ \\ =\dfrac{4\mathrm{\,\ell og\,}2+2\mathrm{\,\ell og\,}3}{3}\\ \\ \\ =\dfrac{4\cdot 0,30+2\cdot 0,48}{3}\\ \\ \\ =\dfrac{1,20+0,96}{3}\\ \\ \\ =\dfrac{2,16}{3}\\ \\ \\ =0,72$