Re-Defining Log: Can We Make it Intuitive?

[tgt]

Would it be okay to redefine the complex definition of Log and define it for example C\[0,infinity)?

I guess then you would have Log z = log |z| + i Arg(z)

where -Pi<=Arg(z)<Pi

Everything would work fine?

But then you can't have Log 5 for example which would be very counter unintuitive.

 

[Gib Z]

Why can't you have Log 5? I see no problems with that. You can redefine anything you want - how useful it ends up being in its applications is another question though.

 

[tgt]

Why can't you have Log 5? I see no problems with that. You can redefine anything you want - how useful it ends up being in its applications is another question though.

To have a Log function defined on the complex plane, you need a branch cut somewhere. Exactly where is arbitary right? So what happens if you choose the positive real line? You'd lose Log (r) for r in the positive reals, including Log(5).

 

[Gib Z]

Assuming log(x) is the logarithim function of the reals, and Log (x) is the complex function you wish to define, your definition gives Log (5) = log |5| + i arg ( 5 + 0i) = log 5 + i*0 = log 5.

I must admit I am somewhat confused as to what is actually happening here - what you gave in the original post is the conventional branch anyway - http://en.wikipedia.org/wiki/Complex_logarithm#Log.28z.29_as_the_inverse_of_the_exponential_function

 

[tgt]

Assuming log(x) is the logarithim function of the reals, and Log (x) is the complex function you wish to define, your definition gives Log (5) = log |5| + i arg ( 5 + 0i) = log 5 + i*0 = log 5.

I must admit I am somewhat confused as to what is actually happening here - what you gave in the original post is the conventional branch anyway - http://en.wikipedia.org/wiki/Complex_logarithm#Log.28z.29_as_the_inverse_of_the_exponential_function

It should be assuming log(|x|) is the logarithm of the reals. I'm just saying since the branch cut can be applied anywhere, what happens if we apply it on the real line? Then we don't have Log(5). Then that wouldn't be a good definition would it?