Natural log of a complex number.

In summary, we are evaluating logarithms of complex numbers and expressing the answers in rectangular form. The formula for the log of a complex number is ln z = ln|z| + argz. The difference between ln a and Ln a is that Ln a is the principal value, meaning it is the value with the smallest absolute value and the argument between 0 and 2kπi. Knowing the principal value can help us find the other values.
  • #1
Drain Brain
144
0
Evaluate the following logarithms, expressing the answers in rectangular form

a. $\ln1$, $Ln1$
b. $\ln(3-j4)$, $Ln(3-j4)$

I know that the log of a complex number z is given as

$\ln z=\ln|z|+argz$

but I still don't know how to use this fact to solve the problems above. I'm having a hard time understanding the material that I read about this. please enlighten me.
 
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  • #2
Drain Brain said:
Evaluate the following logarithms, expressing the answers in rectangular form

a. $\ln1$, $Ln1$
b. $\ln(3-j4)$, $Ln(3-j4)$

I know that the log of a complex number z is given as

$\ln z=\ln|z|+argz$

but I still don't know how to use this fact to solve the problems above. I'm having a hard time understanding the material that I read about this. please enlighten me.

Just a Question: what is the difference in the notations $\ln\ a$ and $\text{Ln}\ a$, being a a complex variable? ...

Kind regards

$\chi$ $\sigma$

- - - Updated - - -

Drain Brain said:
Evaluate the following logarithms, expressing the answers in rectangular form$\ln z=\ln|z|+arg z$

What I remember is that, setting $\displaystyle j = \sqrt{-1}$, is $\displaystyle \ln z = \ln |z| + j\ \text{arg}\ z$...

Kind regards

$\chi$ $\sigma$
 
  • #3
Drain Brain said:
Evaluate the following logarithms, expressing the answers in rectangular form

a. $\ln1$, $Ln1$
b. $\ln(3-j4)$, $Ln(3-j4)$

I know that the log of a complex number z is given as

$\ln z=\ln|z|+argz$

but I still don't know how to use this fact to solve the problems above. I'm having a hard time understanding the material that I read about this. please enlighten me.

Surely you can do ln(1) with your eyes closed.

As for the rest, can't you evaluate |3 - 4j| and arg(3 - 4j) ?
 
  • #4
chisigma said:
Just a Question: what is the difference in the notations $\ln\ a$ and $\text{Ln}\ a$, being a a complex variable? ...

Kind regards

$\chi$ $\sigma$

- - - Updated - - -
What I remember is that, setting $\displaystyle j = \sqrt{-1}$, is $\displaystyle \ln z = \ln |z| + j\ \text{arg}\ z$...

Kind regards

$\chi$ $\sigma$

the $Ln z$ is the principal value of $\ln z$
 
  • #5
Drain Brain said:
the $Ln z$ is the principal value of $\ln z$

That means that is $\displaystyle \ln z = \text{Ln}\ z + 2\ k\ \pi\ i$... so that if you know you have the other one automatically... all right?...

Kind regards

$\chi$ $\sigma$
 

What is the natural log of a complex number?

The natural log of a complex number is the logarithm of the complex number to the base of e, where e is the mathematical constant approximately equal to 2.71828. It is denoted by ln(z) or loge(z), where z is the complex number.

What is the difference between ln(z) and loge(z)?

Both ln(z) and loge(z) represent the natural log of a complex number z. The only difference is the notation used, where ln(z) is more commonly used in mathematics while loge(z) is used in physics and engineering.

Can the natural log of a complex number be negative?

Yes, the natural log of a complex number can be negative. This occurs when the complex number has a negative real part and a non-zero imaginary part. In this case, the natural log will have a negative real part and a complex imaginary part.

What is the relationship between the natural log of a complex number and its argument and modulus?

The natural log of a complex number can be expressed in terms of its argument and modulus using the polar form of a complex number. It is given by ln(z) = ln(r) + iθ, where r is the modulus and θ is the argument of the complex number.

How do you calculate the natural log of a complex number?

To calculate the natural log of a complex number, you can use the polar form of the complex number. First, find the modulus and argument of the complex number. Then, use the formula ln(z) = ln(r) + iθ to find the natural log, where ln(r) is the natural log of the modulus and iθ is the complex part of the natural log. Alternatively, you can use a calculator or a software program to calculate the natural log of a complex number.

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