In this video we're going to look at how
to take the logarithm, the natural
logarithm, of a complex number given in
the standard form. So to do it we need to
use the following result which is going
to be something that we haven't seen in
these videos up to this point, but we're
just going to state it, we're not going
to prove it because the proof is a bit
complicated but it's something that
you'll probably see at some point in
your studies if you are studying complex
numbers. So the result is that r times cos
theta plus i sine theta is equal to r
times e to the i theta. So hopefully the
thing on the left is at least something
you recognise, especially if you've
watched the previous videos in this
series, because it's just a complex
number expressed in polar form. So r is
the modulus, theta is the principal
argument and we have r times cos theta
plus i sine theta,
that's just a complex number in polar
form and this equation tells us that
it's equal to re to the i theta. So this
is a very important equation in
mathematics, it comes up a lot and it tells
us a lot, we can use it many different
ways but in this video we're just going
to use it to calculate the logarithm of
a complex number. So I'm not going to
prove it as I say but just take it as
and given that if you have a number in
polar form there's essentially two ways
of writing it. So if you have a number in
polar form you can write it in this way
on the left hand side, which is called
the polar form, or this way on the right
hand side which is called the
exponential form, but the two are equal.
So if you have the modulus and the
argument you can write it in either way.
So let's use this result to look at the
logarithm of a complex number, so if we
take the natural log of z it's equal to
the natural log of a plus bi. So any
complex number can be expressed in the
form a plus bi. So say we have it in that
form
then we know that we can convert it to
the polar form using the method that
we've seen in previous videos. So if you
have it in standard form you can find r
and theta and write it in polar form and
we know that that's equal to re to the
i theta just using the box at the top. So
once we found r and theta by, well
usually we draw a diagram, do some
trigonometry to find the angle theta and
the length r, we can write the
exponential form down. So we have that
log of a plus bi is equal to log of re to
the i theta but now we use the rules of
logs which hopefully you're familiar
with. So if you have the log of two
things multiplied together, so the log of
a b, it's equal to the log of a plus the
log of b. So that's just a standard rule
for logs. So we have that this is equal
to the log of r plus the log of e to the
i theta, but natural log and e are the
inverse of each other, hopefully you've
seen this, they cancel out. So this is
just equal to log r plus i theta. So in
this second term the log and the e have
cancelled out and we're just left with the
power which was i theta. So this is now
in standard form, a plus bi. It
doesn't maybe look like it is but log of
r, since r is just a real number, log of r
is also a real number. So that's our real
part and then our imaginary part b is
just theta because we have theta times i.
So it is possible if you're given a
number in the form a plus bi and you
take the log of it you can write the
answer also in that form by first
finding r and theta, so doing a bit of
maths to find these two numbers which we
use for the polar form, and then writing
the number in its exponential form using
the equation in the top box and then
doing this using the rule of logs to do
these few final lines to get the answer. 
So let's look at a quick example. We want
to write the log of 1 plus i in the form
a plus bi. Well we need to find r and
theta so we've done that in previous videos, we'll do it again quickly in this one,
the solution is we're going to draw a
quick sketch. So where is 1 plus i in my
complex plane. Well, we go 1 along the
real axis and 1 up the imaginary axis
to i. So it's here where these two lines
meet, so this is the number, this black
dot is the number 1 plus i and the
length of this line here is r and this
angle here is theta.
Okay so r is equal to the square root of
1 plus 1, that's just using Pythagoras's
Theorem or just the standard definition
for r it's the square root of a
squared plus b squared which comes from
Pythagoras's Theorem. So that's just
equal to root 2 and then theta hopefully
you can see is just equal to pi by 4,
because it's half of a right angle,
because this is a square we've drawn
here with these dotted lines and the
line that has length r is going across
one corner to the other so it's cutting
the right angle in the corner of the
square in half. So half of a right angle
which is pi by 2, is pi by 4, so theta is
equal to pi by 4. So we don't even need
to do any trigonometry, we don't need to
use sohcahtoa, once we've drawn our
diagram we can just see that theta is pi
by 4. So now let's use this information
on the right hand side to calculate log
of 1 plus i. 1 plus i written in the
exponential form is square root 2 times
e to the power pi over 4i. So that's
just using the top box there on the left
hand side, so we have that and then that
means that the log of 1 plus i is equal
to the log of square root 2e to the
power pi over 4i. But we just said using
this information on the left what this
is equal to, so let's literally just use
the bottom line from the left hand side
box. We said that this is equal to log r plus
i theta. So that's log of root 2 plus pi
over 4i. So I'm just using the
derivation on the left hand side to
write down straight away what log of 1
plus i is. So all you need to do is find
r and theta and we've done that before,
we know what r in theta are defined
to be it's just the r and theta from
the polar form of the complex number.
Once you've found them you can just plug
them in to the equation that comes from
the box on the left-hand side and we get
log r plus i theta. You could if you
wanted to calculate what log of square
root of 2 is as a decimal, you would put it
into your calculator and you would get a
decimal approximation and you could put
pi over 4 into your calculator to get a
decimal approximation for that, but it's
perfectly fine to just leave them in the
form they're in. Log of square root 2 and
pi over 4. So that's our solution, log of
square root 2 plus pi over 4i is the
log of 1 plus i.
