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The natural logarithm of «x». This is the value y such that ey = 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.


Math functions


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 = 210

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] }

See Also


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