Complex Log?

Algebra Level 4

ln ⁡ z = ln ⁡ 85 + i ( arctan ⁡ 7 6 + π ) \large \ln z = \ln{\sqrt{85}}+i{\left(\arctan{\dfrac{7}{6}}+\pi\right)} If z z is a complex number of the form A + B i A+Bi , where i = − 1 i = \sqrt{-1} , and A , B A,B are real numbers, find the solution z z to the equation above. Enter your answer as A + B A+B .


The answer is -13.

Akeel Howell by Akeel Howell , 22, USA

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1 solution

Guilherme Niedu
Dec 23, 2016

l n ( z ) = l n ( 85 ) + i ⋅ a t a n ( 7 6 ) + i ⋅ π ln(z) = ln(\sqrt{85}) + i \cdot atan(\frac76) + i\cdot\pi

z = e l n ( 85 ) ⋅ e i ⋅ a t a n ( 7 6 ) ⋅ e i ⋅ π z = e^{ln(\sqrt{85})} \cdot e^{i \cdot atan(\frac76)} \cdot e^{i\cdot \pi}

z = 85 ⋅ ( c o s ( a t a n ( 7 6 ) ) + i ⋅ s i n ( a t a n ( 7 6 ) ) ) ⋅ − 1 z = \sqrt{85} \cdot (cos(atan(\frac76)) + i\cdot sin(atan(\frac76)) ) \cdot -1

z = − 85 ⋅ ( 6 85 + i ⋅ 7 85 ) z = -\sqrt{85} \cdot (\frac{6}{\sqrt{85}} + i\cdot \frac{7}{\sqrt{85}} )

z = − 6 − 7 i z = -6 - 7i

A = − 6 , B = − 7 , A+B = -13 \color{#3D99F6} A = -6, B = -7, \fbox{A+B = -13}

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