Difference between revisions of "Ln"
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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)</code>→-3. | + | :<code>Ln(-1)</code> → -3.142j '' Note: when [[EnableComplexNumbers]] is 1'' |
| − | :<code>Ln(2.71828j)</code>→1+1.571j ''Note: [[ImPart]] is <math>\pi/2</math>'' | + | :<code>Ln(2.71828j)</code> → 1+1.571j ''Note: [[ImPart]] is <math>\pi/2</math>'' |
= See Also = | = See Also = | ||
Revision as of 19:40, 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 }[/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 1Ln(2.71828j)→ 1+1.571j Note: ImPart is [math]\displaystyle{ \pi/2 }[/math]
See Also
- LogTen(x)
- Exp(x)
- Complex Numbers
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