Complexities of Logarithm

[etbassoon]

I have two questions about complex logarithms.

First
(-3)^3 = -27, but in Mathematica I get Log[-3,-27]=(i pi + Log[27])/(i pi + Log[3]) instead of 3.

Second
Ln(x+iy) = Ln[sqrt(x^2+y^2)] + i arg(x+iy), so that implies that for x>1, Ln(x^i) = i Ln(x) , Ln|x^i|=0 and arg(x^i)= Ln(x) which does not sit well with me.

Any thoughts?

 

[etbassoon]

I I think I have resolved the second problem.
x^i = exp(i log(x)), so |x^i| = 1, and Ln(|x^i|) = 0,
The arg(x^i) = arctan(Im[x^i]/Re[x^i]) = arctan(Sin[ln(x)]/Cos[ln(x)]) = arctan(tan(ln(x)))=ln(x)

So that's cool, but question 1 has not been resolved.