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Question Number 65971 by gunawan last updated on 07/Aug/19

$$\mathrm{If}\frac{a^2}{b^2}=12\mathrm{then}\log \left({}^3\sqrt{\frac{b}{a}}\right)=..$$$$a.-2$$$$b.-1$$$$c.0$$$$d.1$$$$e.2$$

Answered by MJS last updated on 07/Aug/19

$$\ln \sqrt[3]{\sqrt{12}}=\ln \sqrt[6]{12}=\frac{\ln 12}{6}=\frac{\ln 2}{3}+\frac{\ln 3}{6}$$

Answered by kaivan.ahmadi last updated on 07/Aug/19

$$\frac{a^2}{b^2}=12\Rightarrow {\left(\frac{b}{a}\right)}^2=\frac{1}{12}\Rightarrow \frac{b}{a}=\sqrt{\frac{1}{12}}\Rightarrow$$$$\sqrt[3]{\frac{b}{a}}=\sqrt[6]{\frac{1}{12}}={\left(\frac{1}{12}\right)}^{\frac{1}{6}}={12}^{-\frac{1}{6}}\Rightarrow$$$$log\left(\sqrt[3]{\frac{b}{a}}\right)=log{12}^{-\frac{1}{6}}=-\frac{1}{6}log12=-{log}_{6}12$$$$=-({log}_{6}6+{log}_{6}2)=-1-{log}_{6}2$$$$$$

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