# Ln

## Ln(x)

The natural logarithm of «x». This is the value *y* such that *e ^{y} = Exp(y) = x*, where e=2.718281828459045 is Euler's number.

«x» must be non-negative when complex numbers are not enabled or a warning will be issued. If the warning is ignored, or Show Result Warnings is off, the result is NaN. When complex numbers are enabled, a negative «x» results in a complex number.

## Library

Math functions

## Examples

`Ln(1) → 0`

`Ln(2) → 0.6931471805599453`

`Ln(2.718) → 0.999896315728952`

`Ln(1/2.718) → -0.999896315728952`

`Ln(0) → -INF`

`Ln(-1) → NaN { With Warning:`

*Logarithm of a non-positive number*}

## Base b Logarithms

The base-*b* logarithm of «x» is given by:

`Ln(x) / Ln(b)`

For example:

`Ln(1024) / Ln(2) → 10`

is the base-2 logarithm of 1024, since `1024 = 2`

^{10}

## Complex numbers

When «x» is negative or complex, the result of Ln(x) is a complex number. If you want Ln to return a complex number for a negative parameter, you must set the system variable EnableComplexNumbers to 1, otherwise a warning is issued with a result of NaN. To set EnableComplexNumbers, see enabling complex numbers.

The value of the imaginary part can be interpreted as being in radians.

A complex number can be written in polar form as [math]\displaystyle{ r e^{\theta j} }[/math]. Thus, [math]\displaystyle{ \ln x = \ln r + \theta j }[/math]. In other words, the real part of the result is the log magnitude, and the imaginary part is the phasor angle, [math]\displaystyle{ \theta }[/math], expressed in radians and in [math]\displaystyle{ [-\pi,\pi) }[/math].

`Ln(-1) → -3.142j { When EnableComplexNumbers is 1 }`

`Ln(2.71828j) → 1+1.571j { ImPart is [math]\displaystyle{ \pi/2 }[/math] }`

Enable comment auto-refresher