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Question Number 95968 by i jagooll last updated on 29/May/20

$$3^{\frac{\log {}_3\left(2\right)+\log {}_3\left(3\log {}_{\frac{1}{3}}\left(\cot \frac{\pi }{3}\right)\right)}{\log {}_{\pi }\left(3\right).\left(\log {}_2\left(\pi \right)\right)}?}$$

Answered by john santu last updated on 29/May/20

$$\Rightarrow \log {}_3\left(2\right)+\log {}_3(3\log {}_3\left(\tan \left(\frac{\pi }{3}\right)\right)=$$$$\log {}_3\left(2\right)+\log {}_3\left(3\log {}_3\left(\sqrt{3}\right)\right)=$$$$\log {}_3\left(2\times \frac{3}{2}\right)=1$$$$\Rightarrow \frac{1}{\log {}_{\pi }\left(3\right).\log {}_2\left(\pi \right)}=\frac{1}{\log {}_2\left(3\right)}$$$$\mathrm{so}{3}^{\log {}_3\left(2\right)}=2.$$

Commented by 1549442205 last updated on 29/May/20

$$\mathrm{your}\mathrm{answer}\mathrm{is}\mathrm{false},\mathrm{numerator}\mathrm{equal}$$$$\mathrm{to}1!$$

Commented by 1549442205 last updated on 29/May/20

$${\log }_3\left(2\right)+{\log }_3\left({3\log }_{\frac{1}{3}}\left(\cot \frac{\pi }{3}\right)\right)=$$$${\log }_3\left(2\right)+{\log }_3\left({3\log }_{\frac{1}{3}}{\left(\frac{1}{3}\right)}^{\frac{1}{2}}\right)={\log }_3\left(2\right)+$$$${\log }_3(3.\frac{1}{2})={\log }_3\left(2\right)+\mathrm{lo}\underset{3}{\mathrm{g}3}-{\log }_3\left(2\right)=1$$$$$$$$$$

Commented by i jagooll last updated on 29/May/20

$$\mathrm{yes}...\mathrm{thank}\mathrm{you}$$

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