It is stated that:-
$\log_b(n)$ is undefined, if $n < 0 $.
However,
$\log_i(-1) = 2$, $i$ represents the imaginary number.
Here, $-1 <0$, but the operation is well defined.
My question:-
Is the logarithm of a negative number well defined only if base is an imaginary number?
Short answer: No. The logarithm is defined for all complex numbers not equal to zero.
In complex analysis, the definition of $\log z$ where $z$ is a complex number not equal to zero, is $\log z=\log |z| +i\arg z$, where $\arg z$ gives the angle in the complex plane. Clearly, this reduces to the regular definition for $z>0$.
So, $\log_i (-1)=\log(-1)/\log (i)$; using our formula, $\log(-1)=\log(1)+i\pi$, and $\log(i)=\log(1)+i\pi/2$, giving us your result.
