Logarithm of Complex Numbers

The logarithm of Complex Numbers is another form of expressing complex numbers. While there are several ways to write out complex numbers, there are times when students may be required to write their complex numbers in the logarithm form. As expected, the Log form of the complex numbers employs logarithm processes. More specifically, the natural logarithm is given as ln with a base of e, which is commonly called the exponential symbol. To understand the logarithm of complex numbers properly, students will need to have basic knowledge of the natural logarithm. The good thing, though, is that this piece will outline some of the most important Logarithm patterns because of how their relevance to finding the Log form of complex numbers.

Some Relevant Logarithm Rules

There are some simple Logarithm rules that you should know so that they can be effectively tackled when they come up during complex number considerations. They include

ln(ab) = ln(a) + ln(b)

ln(x/y) = ln(a) – ln(b)

ln(a^b) = bIn(a)

ln(e) = 1

ln(1) = 0

ln(1/x) = – ln(x)

The common logarithm rules above can be interchanged anyways, which also stands when dealing with complex numbers in the logarithm form.

One of the most important options is the ln(e) = 1. The ln is the natural logarithm and is the inverse operation of the exponential term e so. Multiplying a function by its inverse will be 1 because it is like dividing a function by itself. Other outlined common terms are Justas important as well

Logarithm of Complex Numbers Formula

Z = a + bi = r(cosϴ + isinϴ) = re^iϴ

Now, after the first = sign, the rectangular form of the complex number is shown. After that, the polar form of the complex number follows and finally, the Euler form of the complex number follows.

Our concern will be in respect of the Euler form of complex number making

Z = re^iϴ ………… (1)

We will obtain the logarithm formula using the above eqn (1)

Now we will take the in of both sides in eqn(1)

lnZ = lnre^iϴ

now our first common pattern of the ln shows ln(ab) = lna + lnb. As such,

lnZ = lnr + lne^iϴ

lnZ = lnr + iϴlne (since lna^b = bIna)

lnZ = lnr + iϴ (since lne = 1)

eqn (2) is known as the Principal Value of logarithm and is one of the types of the logarithm of complex numbers. There is a general equation of logarithm of z, which is gotten from the principal value of lnz

lnZ = lnr + iϴ

Now while

it is also multi-valued, and for the general natural logarithm of z, it will be given as

ϴ = ϴ + 2xπ where x is an integer.

This will make the Principal value change to

 LnZ = lnr + i(ϴ + 2xπ) ……. (3)

This is the general value of the logarithm of z. The Ln in general, is different as the L is in uppercase while the one for the principal value is lowercase l.

LnZ = lnr + i(ϴ + 2xπ)

The eqn (3) is the General logarithm value of z

Now any of the two logarithms of complex numbers of z can be used depending on the choice that students go for.

The principal logarithm of a complex number formula

General Logarithm of a complex number formula

Now there are some important points to note

Point 1: formerly the a and b in the

are expressed in their absolute value such that even a negative a or b will be assumed to be positive when computing the argument. However, this is not the case in the Logarithm form of complex numbers. If a and b are negative, then they will be considered as such when computing the

Point 2: Also, for the Principal Form of Complex numbers, – π ϴ  π. As such, ϴ is mostly 0 for the principal logarithm form of a complex number

How to Calculate Complex Numbers expressed in Logarithm Forms?

Step 1: Determine which Logarithm form of complex Number formula to be used from the problem given

Step 2: Follow normal Complex Number computations to get an answer

Several Examples of Log form of Complex Number

Example 1

Calculate Ln(-i) and find x = 2, 3, and 4

Note: Keep calculations in radian format

Solution

Based on the above problem, it is obvious that the general formula will be used here, as is evident in the capital L. Also, you should use your calculator set to radian

Now to know the a and b, we have to arrange z

Since z = -i, it means it is actually z = 0 + -1i

Where a = 0, b = – 1

As such;

LnZ = ½(0) + i(π/2 + 2xπ)

LnZ = i(π/2 + 2xπ)

Now we will substitute x = 2, 3, or 4 to get three answers

For x = 2

LnZ = i(π/2 + 2(2)π) = i( π/2 + 4π) = 14.137i

For x = 3

LnZ = i(π/2 + 2(3)π) = i( π/2 + 6π) = 20.42i

For x = 4

LnZ = i(π/2 + 2(4)π) = i( π/2 + 8π) = 26.70i

Example 2:

Now if we have to find ln(-i), we will follow a similar process.

Solution

lnZ = ½(0) + i(π/2)

lnZ = iπ/2

lnZ = 1.571i

Example 3

Find ln(3 + 4i)

Solution

Z = 3 + 4i where a = 3 and b = 4

lnZ = ½(3.2189) + i(0.927)

lnZ = 1.6095 + i(0.927)

lnZ = 1.6095 + 0.927i

The logarithm of complex numbers may be considered to be a bit complicated. However, since it general features π, then it is best for students to have their calculator in radian mode. This piece has generally considered everything about the log form of complex numbers.

Conclusion

This piece has considered the Logarithm o complex numbers. As can be clearly seen, complex numbers in the logarithm form are not any different from other forms of complex numbers. The study of the logarithm brings us close to the finalization of the complex number series that has spanned through the rectangular form of complex numbers, Polar Form of complex numbers, De Moivre of complex numbers and Euler form of complex numbers.

This piece considered three examples to highlight exactly how the logarithm form of complex numbers can be calculated. In general, the computations is best expressed with the calculator in radian form as that is the acceptable way of computing equations and formula having π