Difference between revisions of "Ln"
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== Ln(x) == | == Ln(x) == | ||
| − | The natural logarithm of «x». This is the value ''y'' such that ''e<sup>y</sup> = [[Exp]](y) = x'', where | + | The natural logarithm of «x». This is the value ''y'' such that ''e<sup>y</sup> = [[Exp]](y) = x'', where e2.718281828459045 is Euler's number. |
«x» must be non-negative when [[EnableComplexNumbers|complex numbers are not enabled]] or a warning will be issued. If the warning is ignored, or [[Preferences|Show Result Warnings]] is off, the result is [[NaN]]. When [[EnableComplexNumbers|complex numbers are enabled]], a negative «x» results in a complex number. | «x» must be non-negative when [[EnableComplexNumbers|complex numbers are not enabled]] or a warning will be issued. If the warning is ignored, or [[Preferences|Show Result Warnings]] is off, the result is [[NaN]]. When [[EnableComplexNumbers|complex numbers are enabled]], a negative «x» results in a complex number. | ||
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A complex number can be written in polar form as <math>r e^{\theta j}</math>. Thus, <math>\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>\theta</math>, expressed in radians and in <math>[-\pi,\pi)</math>. | A complex number can be written in polar form as <math>r e^{\theta j}</math>. Thus, <math>\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>\theta</math>, expressed in radians and in <math>[-\pi,\pi)</math>. | ||
| − | :<code>Ln(-1) → -3.142j</code> '' | + | :<code>Ln(-1) → -3.142j</code> {''Note'': when [[EnableComplexNumbers]] is 1 } |
| − | :<code>Ln(2.71828j) → 1+1.571j</code> | + | :<code>Ln(2.71828j) → 1+1.571j</code> { ''Note: [[ImPart]] is <math>\pi/2</math> } |
== See Also == | == See Also == | ||
| − | * [[LogTen]] | + | * [[LogTen]] |
| − | * [[Exp]] | + | * [[Exp]] |
| + | * [[EnableComplexNumbers]] | ||
* [[Complex Numbers]] | * [[Complex Numbers]] | ||
Revision as of 02:04, 19 January 2016
Ln(x)
The natural logarithm of «x». This is the value y such that ey = Exp(y) = x, where e2.718281828459045 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) → 0Ln(2) → 0.6931471805599453Ln(2.718) → 0.999896315728952Ln(1/2.718) → -0.999896315728952Ln(0) → -INFLn(-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{Note: when EnableComplexNumbers is 1 }Ln(2.71828j) → 1+1.571j{ Note: ImPart is [math]\displaystyle{ \pi/2 }[/math] }
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