How Does the Principal Argument Affect the Solution of Log(exp(10i))?

Thread starter NewtonianAlch
Start date Apr 14, 2012
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Complex Logarithms

In summary, a complex logarithm is a mathematical function used to express a complex number in terms of its logarithm. It differs from real logarithms in that it operates on complex numbers and has both a real and imaginary component. Some properties of complex logarithms include the sum and difference rules, as well as the power rule. In science, complex logarithms are used in fields such as physics, engineering, and mathematics for solving equations and modeling real-world phenomena. Some common applications of complex logarithms include signal processing, image and video compression, AC circuit analysis, population modeling, and calculating the decibel scale.

Apr 14, 2012

NewtonianAlch

Homework Statement

Log(exp(10i))

The Attempt at a Solution

Since we want to get this in terms of the principal argument, would it matter if we ultimately wrote i(10 - 3[itex]\pi[/itex]) or i(10 - 4[itex]\pi[/itex]), since they are both in range?

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Apr 14, 2012

jambaugh

Science Advisor

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Yes it matters. i(10 - 3pi) is not a full cycle difference, so not an equivalent angle.

Apr 14, 2012

NewtonianAlch

Thanks!

1. What is a complex logarithm?

A complex logarithm is a mathematical function that is used to express a complex number in terms of its logarithm. It is defined as the inverse of the exponential function and is represented as log_b(z) where z is a complex number and b is the base of the logarithm.

2. How do complex logarithms differ from real logarithms?

Complex logarithms operate on complex numbers, whereas real logarithms only operate on positive real numbers. Complex logarithms have two components - a real part and an imaginary part - while real logarithms do not have an imaginary part.

3. What are the properties of complex logarithms?

Some properties of complex logarithms include:

The logarithm of a product is equal to the sum of the logarithms of each factor.
The logarithm of a quotient is equal to the difference of the logarithms of each term.
The logarithm of a power is equal to the exponent multiplied by the logarithm of the base.
The logarithm of a complex number with a negative real part has a principal value, which is the logarithm of the absolute value of the complex number plus the imaginary part of the complex number.

4. How are complex logarithms used in science?

Complex logarithms are used in various fields of science, including physics, engineering, and mathematics. They are especially useful in solving complex equations and modeling real-world phenomena, such as electrical circuits and population growth.