Matemática, perguntado por gelinhoxgames, 9 meses atrás

Considerando log 2 = 0,3010 e log 3 = 0,4771, calcule:
a) log6 4
log √6
log3^√12
log20
log 0,0002

gelinhoxgames: c) é log ∛12

Soluções para a tarefa

Respondido por Nasgovaskov

Após os cálculos em cada item envolvendo logaritmo, temos como resultado aproximado:

a) 0,773
b) 0,389
c) 0,359
d) 1,301
e) - 3,699

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Considerando log 2 = 0,3010 e log 3 = 0,4771, vamos determinar o valor dos logaritmos em cada item, usando algumas das propriedades dos logaritmos, que deixei em anexo.

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Letra a)

$\begin{array}{l}\sf log\:\!_6\,4=\dfrac{log\,4}{log\,6}\\\\\sf log\:\!_6\,4=\dfrac{log\,2^2}{log\,2\cdot3}\\\\\sf log\:\!_6\,4=\dfrac{2\cdot log\,2}{log\,2+log\,3}\\\\\sf log\:\!_6\,4=\dfrac{2\cdot0,3010}{0,3010+0,4771}\\\\\sf log\:\!_6\,4=\dfrac{0,602}{0,7781}\\\\\!\boldsymbol{\boxed{\sf log\:\!_6\,4\approx0,773}}\end{array}$

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Letra b)

$\begin{array}{l}\sf log\,\sqrt{\:6~}=log\,6^{\,1/2}\\\\\sf log\,\sqrt{\:6~}=\dfrac{~1~}{2}\cdot log\,6\\\\\sf log\,\sqrt{\:6~}=\dfrac{~1~}{2}log\,2\cdot3\\\\\sf log\,\sqrt{\:6~}=\dfrac{~1~}{2}\cdot\big(log\,2+log\,3\big)\\\\\sf log\,\sqrt{\:6~}=\dfrac{~1~}{2}\cdot\big(0,3010+0,4771\big)\\\\\sf log\,\sqrt{\:6~}=\dfrac{~1~}{2}\cdot0,7781\\\\\sf log\,\sqrt{\:6~}=\dfrac{~0,7781~}{2}\\\\\!\boldsymbol{\boxed{\sf log\,\sqrt{\:6~}\approx0,389}}\end{array}$

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Letra c)

$\begin{array}{l}\sf log\,\sqrt[\sf3]{\sf\:12~}=log\,12^{\,1/3}\\\\\sf log\,\sqrt[\sf3]{\sf\:12~}=\dfrac{~1~}{3}\cdot log\,12\\\\\sf log\,\sqrt[\sf3]{\sf\:12~}=\dfrac{~1~}{3}log\,2^2\cdot3\\\\\sf log\,\sqrt[\sf3]{\sf\:12~}=\dfrac{~1~}{3}\cdot\big(log\,2^2+log\,3\big)\\\\\sf log\,\sqrt[\sf3]{\sf\:12~}=\dfrac{~1~}{3}\cdot\big(2\cdot log\,2+log\,3\big)\\\\\sf log\,\sqrt[\sf3]{\sf\:12~}=\dfrac{~1~}{3}\cdot\big(2\cdot0,3010+0,4771\big)\end{array}$

$\begin{array}{l}\sf log\,\sqrt[\sf3]{\sf\:12~}=\dfrac{~1~}{3}\cdot\big(0,602+0,4771\big)\\\\\sf log\,\sqrt[\sf3]{\sf\:12~}=\dfrac{~1~}{3}\cdot1,0791\\\\\sf log\,\sqrt[\sf3]{\sf\:12~}=\dfrac{~1,0791~}{3}\\\\\!\boldsymbol{\boxed{\sf log\,\sqrt[\sf3]{\sf\:12~}\approx0,359}}\end{array}$

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Letra d)

$\begin{array}{l}\sf log\,20=log\,2^2\cdot5\\\\\sf log\,20=log\,2^2+log\,5\\\\\sf log\,20=2\cdot log\,2+log\,\dfrac{10}{2}\\\\\sf log\,20=2log\,2+log\,10-log\,2\\\\\sf log\,20=2\cdot0,3010+1-0,3010\\\\\sf log\,20=0,602+0,699\\\\\!\boldsymbol{\boxed{\sf log\,20\approx1,301}}\end{array}$

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Letra e)

$\begin{array}{l}\sf log\,0,\!0002=log\,\dfrac{2}{10000}\\\\\sf log\,0,\!0002=log\,2-log\,10000\\\\\sf log\,0,\!0002=log\,2-log\,10^4\\\\\sf log\,0,\!0002=log\,2-4\cdot log\,10\\\\\sf log\,0,\!0002=0,3010-4\cdot1\\\\\sf log\,0,\!0002=0,3010-4\\\\\!\boldsymbol{\boxed{\sf log\,0,\!0002\approx-\,3,699}}\end{array}$