Logarithm of a negative number

It is often said that we cannot take logarithm of a negative number. But wait! Suppose we can use complex numbers.

Euler formula

We begin with the Euler formula:

and put q = p, we get :

Therefore,

ln (– 1) = ip

If a > 0, then

ln (– a) = ln [a(– 1)] = ln a + ln (– 1) = ln a + ip

So, ln (– 2) = ln 2 + ip » 0.69315 + 3.1416 i

and 

 

Logarithm of a complex number

By putting q = p/2 in the Euler formula, we get

Therefore

 

If z = b i, which is a purely imaginary number,

then

 

Finally, for the polar form of a complex number,

,

we get

 

Πηγή