# 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 0 1 2 3 4 6 8 10 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.