Logarithm of complex number
Calculator for calculating the natural logarithm (log) of a complex number
The calculator on this page returns the natural logarithm (base e) of a specified complex number. To calculate, enter a complex number and then click the Calculate button.
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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.
More complex functions
Absolute value (abs) • Angle • Conjugate • Division • Exponent • Logarithm to base 10 • Multiplication • Natural logarithm • Polarform • Power • Root • Reciprocal • Square root •Cosh • Sinh • Tanh •
Acos • Asin • Atan • Cos • Sin • Tan •
Airy function • Derivative Airy function •
Bessel-I • Bessel-Ie • Bessel-J • Bessel-Je • Bessel-K • Bessel-Ke • Bessel-Y • Bessel-Ye •
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