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deceptie3j

deceptie3j

Answered question

2022-06-05

general formula for log x ( y ) when y is negative
I'm looking for a general formula for solving a problem of the form log x ( y ) when y < 0. It seems like the formula is ln ( | y | ) + π i ln ( x ) , but I would like to know how this is derived, or a proof for it. I got this this from looking at the answer of wolfram alpha, so this formula may not be correct.

Answer & Explanation

iceniessyoy

iceniessyoy

Beginner2022-06-06Added 27 answers

The complex logarithm function is defined as
Log z := log | z | + i arg z , 0 z C
with log the usual, real logarithm.
But, as we know, the argument of a complex number is defined only up to an integer multiple of 2 π , so the above definition in fact gives us infinite possibilities.
Another fact, way more advanced, is that z = 0 is an acute problem here, and every time we "go around" the origin we add (substract) a multiple of 2 π to the number's argument, so in order to have a more or less nice logarithm function we must choose a complex "branch" for the function, and in this case this means to choose a chunck of the complex plane that we're going to "throw away" so that in the remaining domain the function's well and nicely defined. The main branch here is the non-negative real axis R := { z C ; Re z 0 } , and in C R the function is well defined, continuous and all.
Please do observe then that Log y := log | y | , for any real y

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