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

Complex number +  i
Decimal places

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 \(Im\).

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


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

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

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

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

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