Everything about The Natural Logarithm totally explained
The
natural logarithm, formerly known as the
hyperbolic logarithm, is the
logarithm to the
base e, where
e is an
irrational constant approximately equal to 2.718
281828459. In simple terms, the natural logarithm of a number
x is the power to which
e would have to be raised to equal
x — for example the natural log of
e itself is 1 because
e1 =
e, while the natural logarithm of 1 would be 0, since
e0 = 1. The natural logarithm can be defined for all positive
real numbers
x as the
area under the curve y = 1/
t from 1 to
x, and can also be defined for non-zero
complex numbers as explained below.
The natural logarithm function can also be defined as the
inverse function of the
exponential function, leading to the identities:
»
with
m chosen so that
p bits of precision is attained. In fact, if this method is used, Newton inversion of the natural logarithm may conversely be used to calculate the exponential function efficiently. (The constants ln 2 and
π can be pre-computed to the desired precision using any of several known quickly converging series.)
Computational complexity
The
computational complexity of computing the natural logarithm (using the arithmetic-geometric mean) is O(
M(
n) ln
n). Here
n is the number of digits of precision at which the natural logarithm is to be evaluated and
M(
n) is the computational complexity of multiplying two
n-digit numbers.
Complex logarithms
The exponential function can be extended to a function which gives a
complex number as
ex for any arbitrary complex number
x; simply use the infinite series with
x complex. This exponential function can be inverted to form a complex logarithm that exhibits most of the properties of the ordinary logarithm. There are two difficulties involved: no
x has
ex = 0; and it turns out that
e2πi = 1 =
e0. Since the multiplicative property still works for the complex exponential function,
ez =
ez+2nπi, for all complex
z and integers
n.
So the logarithm can't be defined for the whole
complex plane, and even then it's multi-valued – any complex logarithm can be changed into an "equivalent" logarithm by adding any integer multiple of 2
πi at will. The complex logarithm can only be single-valued on the
cut plane. For example, ln
i = 1/2
πi or 5/2
πi or −3/2
πi, etc.; and although
i4 = 1, 4 log
i can be defined as 2
πi, or 10
πi or −6
πi, and so on.
Image:NaturalLogarithmRe.png| z = Re(ln(x+iy))
Image:NaturalLogarithmIm.png| z = |Im(ln(x+iy))|
Image:NaturalLogarithmAbs.png| z = |ln(x+iy)|
Image:NaturalLogarithmAll.png| Superposition of the previous 3 graphs
Further Information
Get more info on 'Natural Logarithm'.
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