Difference between revisions of "Ln"
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[[Category:Math Functions]] | [[Category:Math Functions]] | ||
| + | [[Category:Functions that operate on complex numbers]] | ||
[[Category:Doc Status D]] <!-- For Lumina use, do not change --> | [[Category:Doc Status D]] <!-- For Lumina use, do not change --> | ||
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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. | 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 or a warning will be issued. If the warning is ignored, or [[Preferences|Show Result Warnings]] is off, the result is [[NaN]]. | + | «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. |
= Library = | = Library = | ||
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:Ln(1024) / Ln(2) → 10 | :Ln(1024) / Ln(2) → 10 | ||
is the base-2 logarithm of 1024, since 1024 = 2<sup>10</sup> | is the base-2 logarithm of 1024, since 1024 = 2<sup>10</sup> | ||
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| + | = Complex numbers = | ||
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| + | When «x» is negative or complex, the result of <code>[[Ln]](x)</code> 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 [[EnaleComplexNumbers|enabling complex numbers]]. | ||
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| + | The value of the imaginary part can be interpreted as being in radians. | ||
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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. | ||
= See Also = | = See Also = | ||
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* [[LogTen]](x) | * [[LogTen]](x) | ||
* [[Exp]](x) | * [[Exp]](x) | ||
| + | * [[ComplexNumbers]] | ||
<comments /> | <comments /> | ||
Revision as of 19:24, 15 April 2013
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) → 0
- Ln(2) → 0.6931471805599453
- Ln(2.718) → 0.999896315728952
- Ln(1/2.718) → -0.999896315728952
- Ln(0) &rarr -INF
- Ln(-1) &rarr 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\lt math\gt . In other words, the real part of the result is the log magnitude, and the imaginary part is the phasor angle, \lt math\gt \theta }[/math], expressed in radians.
See Also
- LogTen(x)
- Exp(x)
- ComplexNumbers
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