## ln(i) using Euler's Equation

Problem: Using Euler's Equation, evaluate $$\ln i$$. Euler's Equation is given by: $e^{ix}=\cos x +i\sin x$ (a proof of this equation can be found here)

Solution:

Consider what we get if we $$ln$$ both sides of Euler's Equation:

$\ln\left(e^{ix}\right)=\ln(\cos x +i\sin x)$

$ix=\ln(\cos x +i\sin x)$

Now we want $$\ln i$$ , so notice that we can get that on the RHS if we let $$x=\frac{\pi}{2}$$:

$i\frac{\pi}{2}=\ln \left(\cos \frac{\pi}{2} +i\sin \frac{\pi}{2}\right)$

$i\frac{\pi}{2}=\ln(i)$

$\therefore \ln i = \frac{\pi i}{2}$