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Question Number 174507 by Mastermind last updated on 02/Aug/22

Determine the numerical value of the   following expression without the use  of a calculator  log[log(3)∙(log(2)∙((((√3)−2sin((π/3)))/(π^3 +1))+1))−log(2)log(3)+(−1)^(100) ]    Mastermind

$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{numerical}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{following}\:\mathrm{expression}\:\mathrm{without}\:\mathrm{the}\:\mathrm{use} \\ $$$$\mathrm{of}\:\mathrm{a}\:\mathrm{calculator} \\ $$$$\mathrm{log}\left[\mathrm{log}\left(\mathrm{3}\right)\centerdot\left(\mathrm{log}\left(\mathrm{2}\right)\centerdot\left(\frac{\sqrt{\mathrm{3}}−\mathrm{2sin}\left(\frac{\pi}{\mathrm{3}}\right)}{\pi^{\mathrm{3}} +\mathrm{1}}+\mathrm{1}\right)\right)−\mathrm{log}\left(\mathrm{2}\right)\mathrm{log}\left(\mathrm{3}\right)+\left(−\mathrm{1}\right)^{\mathrm{100}} \right] \\ $$$$ \\ $$$$\mathrm{Mastermind} \\ $$

Answered by a.lgnaoui last updated on 03/Aug/22

log[log(3).(log(2).((((√3) −2((√3)/2))/(π^3 +1))+1))−log(2)log(3)+1]  =log[log(3).(log(2)−log(2)log(3)+1]  =log(1)=0

$$\mathrm{log}\left[\mathrm{log}\left(\mathrm{3}\right).\left(\mathrm{log}\left(\mathrm{2}\right).\left(\frac{\sqrt{\mathrm{3}}\:−\mathrm{2}\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}{\pi^{\mathrm{3}} +\mathrm{1}}+\mathrm{1}\right)\right)−\mathrm{log}\left(\mathrm{2}\right)\mathrm{log}\left(\mathrm{3}\right)+\mathrm{1}\right] \\ $$$$=\mathrm{log}\left[\mathrm{log}\left(\mathrm{3}\right).\left(\mathrm{log}\left(\mathrm{2}\right)−\mathrm{log}\left(\mathrm{2}\right)\mathrm{log}\left(\mathrm{3}\right)+\mathrm{1}\right]\right. \\ $$$$=\mathrm{log}\left(\mathrm{1}\right)=\mathrm{0} \\ $$

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