When can I assume Log means the natural log? I have a problem that says: Express e^(Log(3+4i)) in the form x+iy. However, in class we usually only deal with natural log (this problem is from a textbook). How do I know when I can interpret ''Log'' as the natural log?

Gisselle Hodges

Gisselle Hodges

Answered question

2022-10-16

When can I assume Log means the natural log?
I have a problem that says:
Express e Log ( 3 + 4 i ) in the form x + i y
However, in class we usually only deal with natural log (this problem is from a textbook). How do I know when I can interpret ''Log'' as the natural log?

Answer & Explanation

pawia6g

pawia6g

Beginner2022-10-17Added 14 answers

Since z e z is periodic with period 2 π i, it is not one-to-one, and so its inverse is a "multiple-valued function" (and so not really a "function" as textbooks define that term over the past hundred years or so). Thus
Log ( 3 + 4 i ) = log 3 2 + 4 2 + i ( θ 0 + 2 π n )
and θ 0 + 2 π n is any of the angels in the first quadrant whose tangent is 4 / 3
Exponentiating this is exponentiating any of the complex number that when exponentiated yield 3 + 4 i. Hence
e Log ( 3 + 4 i ) = 3 + 4 i .

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