Complex Logarithms

[00:00:00]So, here what we're going to do is calculate in Cartesian form the log of 5e to the minus 3 pi over 4 minus i3 pi over 4. And we've got a capital L here, so we want the principal logarithm. So, the principal logarithm, we're just looking for one solution. Now, logarithms in complex numbers have infinite solutions. So, with the result of this will be of the format the log of z will give us the logarithm of the modulus of z plus our imaginary component will be the principal argument of that z.

[00:00:35]Now, the principal argument needs to be between minus pi and pi. So, that's what we need to just check for our end at the end.

[00:00:41]So, let's assume log of z, let z be this. So, let's take care of the right-hand side.

[00:00:48]So, we've got the log of the modulus of 5e to the minus i 3 pi over 4.

[00:00:57]And then add on to that our imaginary component, which is the argument of all of that. That's 5e to the minus i 3 pi over 4.

[00:01:08]Okay, first of all, let's take care of this one. Now, here we've got 5 times the exponential of minus i times some number. Now, e times minus i times some number, that is always going to give us a ray of length one. So, the modulus of log of 5 e of this stuff is just going to be five. So, here we've just down to the log of 5.

[00:01:36]And then here, what's our argument here? Well, the argument is just our direction of the ray. So, the length of it is unimportant. So, this five here, we can kind of cancel that out. And then we're left with e to the minus i 3 pi over 4.

[00:01:52]So, that tells us our principal argument is minus 3 pi over 4.

[00:01:58]Now, that is a number that is between minus pi and pi. So, the answer to that is just simply going to be minus So, we've got the i there.

[00:02:08]And then 3 pi over 4.

[00:02:10]That's our principal argument with the capital A. And as it's a minus, we just subtract. And that's our number in the form of a plus i b. In this case, our b would be a negative. And then we are done.