1. The problem asks for the principal value of $\ln i$. 2. Recall that for a complex number $z = re^{i\theta}$, the principal value of the natural logarithm is given by: $$\ln z = \ln r + i\theta$$ where $r = |z|$ and $\theta = \arg(z)$ with $\theta$ in the principal range $(-\pi, \pi]$. 3. For $i$, we have: - Magnitude: $r = |i| = 1$ - Argument: $\theta = \frac{\pi}{2}$ (since $i$ lies on the positive imaginary axis) 4. Substitute these values: $$\ln i = \ln 1 + i \cdot \frac{\pi}{2} = 0 + i \frac{\pi}{2} = i \frac{\pi}{2}$$ 5. Therefore, the principal value of $\ln i$ is: $$\boxed{i \frac{\pi}{2}}$$