The logarithm of a complex number $z=re^{i\theta}$ is defined (I think) as:

$$\log z=\ln |z| + i\arg z=\ln r+i\theta.$$

What is the value of $\log -1$. Following the definition, it is $\log-1=\ln1+i(-\pi)=-i\pi$. Is this correct?

Since the value $i\log i=\log i^i$ is well defined. What does $i^i$ means?

Am I missing something here? 

Thank you for your time.

**EDIT** My question is raised when I started trying to calculate the value of $$\int\_{0}^{\infty}\dfrac{1}{x^2+1}\mathrm{d}x=\int\_{0}^{\infty}\left(\dfrac{i}{2(x+i)}+\dfrac{-i}{2(x-i)}\right)\mathrm{d}x$$

After my calculation I found $$\int\_{0}^{\infty}\dfrac{1}{x^2+1}\mathrm{d}x=i\log-1.$$