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Find-the-value-of-log-2-imaginary-




Question Number 100442 by Dwaipayan Shikari last updated on 26/Jun/20
Find the value of     log(−2)  {imaginary}
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\:\:\:\mathrm{log}\left(−\mathrm{2}\right)\:\:\left\{\mathrm{imaginary}\right\} \\ $$
Commented by mathmax by abdo last updated on 26/Jun/20
you are welcome.
$$\mathrm{you}\:\mathrm{are}\:\mathrm{welcome}. \\ $$
Commented by Dwaipayan Shikari last updated on 26/Jun/20
log(−2)=log(−1)+log(2)                     =log(e^(iπ) )+log(2)                         =log(e^(i(2kπ+π)) )+log(2)                      =iπ(2k+1)+log(2)    {k∈Z}  Is  this a solution???????  I am not sure
$$\mathrm{log}\left(−\mathrm{2}\right)=\mathrm{log}\left(−\mathrm{1}\right)+\mathrm{log}\left(\mathrm{2}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{log}\left(\mathrm{e}^{\mathrm{i}\pi} \right)+\mathrm{log}\left(\mathrm{2}\right)\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{log}\left(\mathrm{e}^{\mathrm{i}\left(\mathrm{2k}\pi+\pi\right)} \right)+\mathrm{log}\left(\mathrm{2}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{i}\pi\left(\mathrm{2k}+\mathrm{1}\right)+\mathrm{log}\left(\mathrm{2}\right)\:\:\:\:\left\{\mathrm{k}\in\mathbb{Z}\right\} \\ $$$$\mathrm{Is}\:\:\mathrm{this}\:\mathrm{a}\:\mathrm{solution}??????? \\ $$$$\mathrm{I}\:\mathrm{am}\:\mathrm{not}\:\mathrm{sure} \\ $$$$ \\ $$
Commented by mr W last updated on 26/Jun/20
i think it′s correct.
$${i}\:{think}\:{it}'{s}\:{correct}. \\ $$
Commented by abdomsup last updated on 26/Jun/20
ln(−2) =ln(−1)+ln(2)  ln(e^(i(π+2kπ)) )+ln(2)  =ln2 +i(2k+1)π    (k∈Z)
$${ln}\left(−\mathrm{2}\right)\:={ln}\left(−\mathrm{1}\right)+{ln}\left(\mathrm{2}\right) \\ $$$${ln}\left({e}^{{i}\left(\pi+\mathrm{2}{k}\pi\right)} \right)+{ln}\left(\mathrm{2}\right) \\ $$$$={ln}\mathrm{2}\:+{i}\left(\mathrm{2}{k}+\mathrm{1}\right)\pi\:\:\:\:\left({k}\in{Z}\right) \\ $$
Commented by Dwaipayan Shikari last updated on 26/Jun/20
Thanking you for your confirmation Mr.W sir and Abdomsup sir
$$\mathrm{Thanking}\:\mathrm{you}\:\mathrm{for}\:\mathrm{your}\:\mathrm{confirmation}\:\mathrm{Mr}.\mathrm{W}\:\mathrm{sir}\:\mathrm{and}\:\mathrm{Abdomsup}\:\mathrm{sir} \\ $$

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