How does the complex logarithm work?

How does the complex logarithm work?



How does the complex logarithm work?

The complex logarithm is a mathematical function that allows you to determine the exponent to which a given base must be raised to obtain a specified complex number. In other words, it solves the following equation: b^x = zWhere b is the basis, x is the unknown exponent and z is the given complex number.

To understand how the complex logarithm works, it is necessary to understand the nature of complex numbers. A complex number can be written in the form a + biWhere a et b are real numbers and i is the imaginary unit. Imaginary unity i is defined as the square root of -1.

The complex logarithm is therefore used to solve equations involving complex numbers. There are several approaches to calculating the complex logarithm, including the natural logarithm method and the complex angle method.

Natural logarithm method:

The natural logarithm method uses the natural logarithm function to calculate the complex logarithm. To find the complex logarithm of a complex number z = a + bi, we use the following formula:

ln(z) = ln(|z|) + i * arg(z)

where | from | is the module of z (the distance between the origin and the point representing z in the complex plane) and arg(z) is the argument of z (the angle between the positive real axis and the vector representing z in the complex plane).

Once ln(z) is calculated, it is possible to find the exponent x in b^x = z using the following formula:

x = ln(z) / ln(b)

Complex angles method:

The complex angle method uses the properties of complex numbers and angles to calculate the complex logarithm. It is based on the following formula:

ln(z) = ln(|z|) + i * arg(z)

where | from | et arg(z) are the same as in the natural logarithm method.

Using the properties of complex angles, it is possible to simplify this formula and express the complex logarithm more concisely. However, this method requires a good understanding of complex numbers and the properties of angles.

It is important to note that the complex logarithm can have several possible values, because there are branch differences. This means that a logarithmic equation cannot have just one unique solution, but rather infinitely many solutions.

Example:

Consider the logarithmic equation i^x = -1. To solve this equation, we use the natural logarithm method:

ln(-1) = ln(|-1|) + i * arg(-1)

ln(-1) = ln(1) + i * (-π/2 + 2πn) for all n

Therefore, the complex logarithm of -1 is i * (-π/2 + 2πn) for all n.



Similar searches:

1. What are the applications of the complex logarithm in science?

The complex logarithm is used in many fields of science, such as physics, engineering, and mathematics. It is used to solve differential equations, analyze electrical circuits, study growth and decay phenomena, model dynamic systems, among others. The applications of the complex logarithm are vast and essential for understanding and solving complex problems in various scientific disciplines.

source: Family Business – Oct 2001 – Page 50

2. What are the properties of the complex logarithm?

The complex logarithm has several important properties. Some of these properties are:

  • The complex logarithm of the multiplication of two complex numbers is equal to the sum of the complex logarithms of these numbers.
  • The complex logarithm of dividing two complex numbers is equal to the difference of the complex logarithms of these numbers.
  • The complex logarithm of the exponentiation of a complex number is equal to the product of the exponent and the complex logarithm of that number.
  • The complex logarithm of the nth root of a complex number is equal to dividing the complex logarithm of that number by n.

These properties of the complex logarithm are similar to those of the real logarithm, but adapted to complex numbers.

source: Backpacker – Mar 2001 – Page 15

3. What is the relationship between complex logarithm and exponential functions?

The complex logarithm is closely related to exponential functions. Indeed, the complex exponential function and the complex logarithm are inverse functions of each other. This means that if b^x = zThen x = log_b(z) and vice versa.

This relationship between the complex logarithm and exponential functions is similar to that of the real logarithm and the real exponential function. It allows you to solve exponential and logarithmic equations involving complex numbers.

source: Backpacker – Mar 2001 – Page 15

Sources used:

Backpacker – Mar 2001 – Page 15 – Google Books Result

Family Business – Oct 2001 – Page 50 – Google Books Result

Last consultation of sources: 2023-08-01

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