Logarithm of a Complex number question?

In summary, the question is asking for the next step in solving a problem involving logarithms and complex numbers. The provided equations and values are used to find the angle and principal value for the given expressions. The next step is to plug in the value for c and double check the angle for the first equation.
  • #1
Mandynash
3
0

Homework Statement



The question is located here http://i51.tinypic.com/2cge9mt.jpg


Homework Equations



My a value is -3
my b value is -3sqrt(2)
my c value is -2.4

The Attempt at a Solution



1) ln(-3 - 3 sqrt(2) i)
= Ln |-3 - 3 sqrt(2) i| + i arg(-3 - 3 sqrt(2) i)
= ln sqrt(27) + i (-3π/4 + 2πk), for any integer k

2) ln(-c) = Ln |-c| + arg(-c) = ln c + i(π + 2πk) for any integer k.

(Principal value occurs for k = 0.)


Managed to get this far but I have no idea what I need to do next! Thanks in advance for the help.
 
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  • #2
For 2) you just have to plug c=2.4.

For 1) you may want to check the angle again. I get [itex] \arccos\left(-\frac{\sqrt{3}}{3}\right) [/itex]
 

1. What is the definition of a logarithm of a complex number?

The logarithm of a complex number is the power to which a base number must be raised to equal the given complex number. It is denoted as logb(z) where b is the base and z is the complex number.

2. How is the logarithm of a complex number calculated?

The logarithm of a complex number can be calculated using the natural logarithm function, ln, on the absolute value of the complex number. It can also be calculated using the logarithm laws for complex numbers, such as log(z1 * z2) = log(z1) + log(z2).

3. Can the logarithm of a complex number have multiple values?

Yes, the logarithm of a complex number can have multiple values. This is because the complex logarithm is a multivalued function, meaning there can be more than one result for a given input. These multiple values are called branches of the complex logarithm.

4. What are the properties of the logarithm of a complex number?

The properties of the logarithm of a complex number include:

  • log(z1 * z2) = log(z1) + log(z2)
  • log(z1 / z2) = log(z1) - log(z2)
  • log(z1n) = n * log(z1)
  • log(z10) = 0
  • log(1) = 0

5. In what fields of science is the logarithm of a complex number used?

The logarithm of a complex number is commonly used in fields such as mathematics, physics, engineering, and computer science. It is especially useful in solving problems involving exponential growth and decay, as well as in analyzing the behavior of signals and systems.

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