Question

Subject:maths
Chapter:logarithms​

Answer 1

Given :

$\sf \bullet \: \: log_{10}(2) = \: 0.3010 \\ \\ \sf \bullet \: \: log_{10}(3) = \: 0.4771$

To find :

$\sf \: log( \sqrt[3]{48} \: \times \: \sqrt{108} \: \div \: 12 \sqrt{6} )$

Identity used :

$\sf \bullet \: \: log( {a}^{m} ) = m \times log(a) \\ \\ \sf \bullet \: \: log(a \times b) \: = \: log(a) + \: log(b) \\ \\ \sf \bullet \: log(a \div b) = \: log(a) - log(b)$

Solution :

First of all we should simplify the terms , inside of log

$\sf \bullet \: \sqrt[3]{48} = \sqrt[3]{(2 \times 2 \times 2 \times 2 \times 3)} \\ \\ \sf \implies \: \sqrt[3]{48} = \sqrt[3]{ {2}^{3} \times 2 \times 3} \\ \\ \sf \implies \: \sqrt[3]{48} = 2 \times ( \sqrt[3]{2 \times 3} )$

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$\bullet \: \: \sqrt{108} \: = \: \sqrt{(2 \times 2 \times 3 \times 3 \times 3)} \\ \\ \implies \: \sqrt{108} = \sqrt{ {2}^{2} \times {3}^{3} } \\ \\ \implies \: \sqrt{108} = (2 \times 3)\sqrt{3}$

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$\bullet \: \: 12 \times \sqrt{6} = 2 \times 2 \times 3 \times \sqrt{2 \times 3}$

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Therefore,

$\sf \: log( \sqrt[3]{48} \: \times \: \sqrt{108} \: \div \: 12 \sqrt{6} ) \\ \\ \sf \implies \: log((2 \times \sqrt[3]{2} \times \sqrt[3]{3} ) \times (2 \times 3 \times \sqrt{3}) \div ( {2}^{2} \times 3 \times \sqrt{2 } \times \sqrt{3} )) \\ \\ \sf \implies \: log( \dfrac{ {2}^{2} \times 3 \times \sqrt[3]{2} \times \sqrt[3]{3} \times \sqrt{3} }{ {2}^{2} \times 3 \times \sqrt{2} \times \sqrt{3} } ) \\ \\ \sf \implies log( \dfrac{ \sqrt[3]{2} \times \sqrt[3]{3} }{ \sqrt{2} } ) \\ \\ \sf \implies \: log( \sqrt[3]{2} ) + log( \sqrt[3]{3} ) - log( \sqrt{2} ) \\ \\ \sf \implies \: \dfrac{1}{3} log(2) \: + \: \dfrac{1}{3} log(3) - \dfrac{1}{2} log(2) \\ \\ \sf \implies \: \dfrac{1}{3} \times 0.3010 \: + \dfrac{1}{3} \times 0.4771 \: - \dfrac{1}{2} 0.3010 \\ \\ \sf \implies \: 0.1003 \: + \:0.1590 \: - \: 0.1505 \\ \\ \sf \implies \bold{0.1088}$

ANSWER : 0.1088