What is ln(sqrt(2)e^{(-pi)/4 i}) ?

The solution to ln(sqrt(2)e^{(-pi)/4 i}) is (ln(2))/2-i pi/4

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simplify ln(sqrt(2)e^{(-pi)/4 i})

simplify

ln (2 e^{\frac{- π}{4} i})

\frac{ln ( 2 )}{2} - i \frac{π}{4}

Solution steps

ln (2 e^{\frac{- π}{4} i})

Apply log rule:

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

ln (2 e^{\frac{- π}{4} i}) = ln (2) + ln (e^{\frac{- π}{4} i})

= ln (2) + ln (e^{i \frac{- π}{4}})

Apply log rule:

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

ln (e^{\frac{- π}{4} i}) = \frac{- π}{4} i ln (e)

= ln (2) + \frac{- π}{4} i ln (e)

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

\frac{- π}{4} i ln (e) = - \frac{πi}{4}

= \frac{1}{2} ln (2) - \frac{πi}{4}

Rewrite

\frac{1}{2} ln (2) - \frac{πi}{4}

in standard complex form:

\frac{ln ( 2 )}{2} - \frac{π}{4} i

= \frac{ln ( 2 )}{2} - \frac{π}{4} i

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What is ln(sqrt(2)e^{(-pi)/4 i}) ?
The solution to ln(sqrt(2)e^{(-pi)/4 i}) is (ln(2))/2-i pi/4