MHB Why is ln(k) a Complex Number When k is a Positive Integer?

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The discussion centers on the nature of the natural logarithm function, specifically why ln(k) is considered a complex number when k is a positive integer. Participants explain that while k is positive, the logarithm can yield complex results due to its definition in the complex plane, particularly involving the Euler's formula. The argument highlights that ln(k) can be expressed as ln(k) + 2πi*n, where n is any integer, indicating the multi-valued nature of logarithms in complex analysis. Clarifications are made regarding the principal value of the logarithm, which is real for positive k but can extend into the complex realm. The conversation emphasizes the importance of understanding logarithmic functions in both real and complex contexts.
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Why ln(k) when k is a possitive integer, ln(k) is a complex number?
 
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Why do you think \(\ln(k)\) is complex?
 

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