What is ln(1/2 e^{ipi}) ?

The solution to ln(1/2 e^{ipi}) is -ln(2)+ipi

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simplify ln(1/2 e^{ipi})

simplify

ln (\frac{1}{2} e^{iπ})

- ln (2) + iπ

Solution steps

ln (\frac{1}{2} e^{iπ})

Apply log rule:

lo g_{c} (ab) = lo g_{c} (a) + lo g_{c} (b)

ln (\frac{1}{2} e^{iπ}) = ln (\frac{1}{2}) + ln (e^{iπ})

= ln (\frac{1}{2}) + ln (e^{iπ})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (e^{iπ}) = iπ ln (e)

= ln (\frac{1}{2}) + iπ ln (e)

Apply log rule:

lo g_{c} (\frac{a}{b}) = lo g_{c} (a) - lo g_{c} (b)

ln (\frac{1}{2}) = ln (1) - ln (2)

= ln (1) - ln (2) + iπ ln (e)

ln (1) = 0

iπ ln (e) = iπ

= 0 - ln (2) + iπ

0 - ln (2) + iπ = - ln (2) + iπ

= - ln (2) + iπ

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