Complex powers/logarithmic spirals

[hoodwink]

When looking at

w^z = e^(z log w)

I understand that adding any integer multiple of (2*pi*i) to log w is equivalent to a full rotation in the complex plane. I don't understand how this step is equivalent to multiplying w^z by e^(z*2*pi*i). Also, I'm missing the significance of this being represented in the complex plane as the intersections of 2 logarithmic spirals. I can see how the first spiral is given by w^z, but the other?

If anyone has a copy handy, my questions arose from looking at pages 96-97 of The Road to Reality by Roger Penrose.

 

[Diffy]

I think it is just rules of exponents from algebra.

(w^z)*e^(z*2*pi*i) = e^(z log w)*e^(z*2*pi*i) = e^(z log w + z*2*pi*i) =
e^(z(log w + 2*pi*i))

Does this help?

 

[hoodwink]

click! thanks for that diffy. hopefully those spirals will start to do the same now...