Logarithm of complex number

Calculator for calculating the natural logarithm (log) of a complex number

Logarithm of a complex number


The function returns the natural logarithm (base e) of a specified complex number.


Logarithm calculator

 Input
Complex number +  i
Decimal places
 Results
Logarithm

Formulas for calculation of the logarithm

In the following description,\(z\) stands for the complex number.
\(x\) stands for the real value \(Re\) and \(y\) for the imaginary value \(Im\).

\(\displaystyle ln(z) = \frac{1}{2} · ln\left(x^2 + y^2\right) + atan\left(\frac{y}{x}\right)\)

Example

\(ln(z) = ln(3+5i)\)

\(\displaystyle Re = \frac{1}{2} · ln\left(3^2 + 5^2\right) = \frac{1}{2} · ln(9 + 25) =1.763\)

\(\displaystyle Im = atan\left(\frac{5}{3}\right) = 1.030\)

\(\displaystyle ln(3+5i) = 1.763+1.030i\)

The result for \(\displaystyle Im \) is given in radians.
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