Our motivation for the definition of the logarithmic function is based on solving the equation $$(1) e^w=z$$ for w, where z is any nonzero complex number. To do this, we note that when z and w are written $$z=re^i\theta (-\pi<\theta<=\pi)$$ and $$w=u+iv$$, equation (1) becomes $$e^u*e^ie^v=re^i\theta$$ This tells us that $$e^u=r$$ and $$v=a\theta+2n\pi$$ where n is any integer. Since the equation $$e^u=r$$ is the same as $$u=lnr$$, it follows that equation (1) is satisfied if and only if w has one of the values $$w=lnr+i(\theta+2n\pi) (n is integer)$$. Thus, we write $$(2) logz=lnr+i(\theta+2n\pi) (n is integer)$$, equation (1) tells us that (3)e^(logz)=z (z=/0)
This is from "complex variables and applications, 9th edition, Brown). I cannot understand (2) equation. Why is it logz, not lnz?