What I've just asked might appear ambiguous and I suppose I could clarify myself with an example: e to the 0 is 1 as it should be, but e to the 2πi is 1 at the same time, so the base e logarithm of 1 is 0 and 2πi simultaneously. What I'm asking is if there's a method in which we can indicate a specific power to pick as an output from a logarithm operation. For instance, if we use ln(1) in an operation, I'm asking for the method in which we can use ln(1) as 2πi by indicating that it should be that number (instead of the default 0).
I have an idea about this, but it's not based on anything official. Basically, I thought that you should be able to indicate logarithms as ln(x): t=y where x is preferably a complex number, and y is the total number of 2π radian turns around the complex plane. For instance, if I say ln($\sqrt{2}$+$\sqrt{2}$i): t=0, its equivalent is π/4 and if I say ln($\sqrt{2}$+$\sqrt{2}$i): t=1, its equivalent is 9π/4 and so on. So what I'm doing here is basically ln(x)+y·2π but without using the irrational numbers.