Natural logarithm
The natural logarithm, formerly known as the hyperbolic logarithm, is the
logarithm to the base
e, where
e is equal to 2.718281828459... . 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 will be explained below.
Encyclopedia
The
natural logarithm, formerly known as the hyperbolic logarithm, is the
logarithm to the base
e, where
e is equal to 2.718281828459... . 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 will be explained below.
Notational conventions
- Mathematicians generally understand either "log" or "ln" to mean loge, i.e., the natural logarithm of x, and write "log10" if the base-10 logarithm of x is intended.
- Engineers, biologists, and some others write only "ln" or "loge" when they mean the natural logarithm of x, and take "log" to mean log10 or, in the context of computing, log2.
- On hand-held calculators the natural logarithm is ln, whereas log is the base-10 logarithm.
See also
logarithms.
The natural logarithm is the inverse of the natural exponential function
This function is the inverse function of the
exponential function:
for all positive
x and
for all real
x.
In other words, the logarithm function is a
bijection from the set of positive real numbers to the set of all real numbers. More precisely it is an isomorphism from the group of positive real numbers under multiplication to the group of real numbers under addition.
Logarithms can be defined to any positive base other than 1, not just
e, and they are always useful for solving equations in which the unknown appears as the exponent of some other quantity.
Reason for being "natural"
Initially, it seems that in a world using base 10 for nearly all calculations, this base would be more "natural" than base
e. The reason we call the ln "natural" is twofold: first, expressions in which the unknown variable appears as the exponent of
e occur
much more often than exponents of 10 , and second, because the natural logarithm can be defined quite easily using a simple integral or
Taylor series--which is not true of other logarithms. Thus, the natural logarithm is more useful in practice.
To put it concretely, consider the problem of differentiating a logarithmic function:
If the base
b is
e then the derivative is 1/
x, and at
x = 1 the slope of the graph is 1.
There are other reasons the natural logarithm is natural: there are a number of simple series involving the natural logarithm, and it often arises in nature. Indeed, Nicholas Mercator first described them as
log naturalis before calculus was even conceived.
Definitions
Formally, log may be defined as the area under the graph of
1/
x from 1 to
a, that is,
This defines a logarithm because it satisfies the fundamental property
of a logarithm:
This can be shown by defining and using the substitution rule of integration as follows:
The number
e can then be defined as the unique real number such that .
Alternatively, if the
exponential function has been defined first using an infinite series, the natural logarithm may be defined as its inverse function, meaning log is that number for which Since the range of the exponential function is all positive real numbers and since the exponential function is strictly increasing, this is well-defined for all positive
x.
Derivative, Taylor series and complex arguments
The
derivative of the natural logarithm is given by
This leads to the
Taylor serieswhich is also known as the Mercator series.
An alternative form for log itself is
The natural logarithm in integration
The natural logarithm allows simple
integration of functions of the form
g =
f '/
f: an
antiderivative of
g is given by ln. This is the case because of the chain rule and the following fact:
In other words,
and
Here is an example in the case of
g = tan:
Letting
f = cos and
f= - sin:
where C is an arbitrary constant of integration.
The natural logarithm can be integrated using integration by parts:
Numerical value
To calculate the numerical value of the natural logarithm of a number, the Taylor series expansion can be rewritten as:
To obtain a better rate of convergence, the following identity can be used.
provided that
y = / and
x > 0.
For log where
x > 1, the closer the value of
x is to 1, the faster the rate of convergence. The identities associated with the logarithm can be leveraged to exploit this:
Such techniques were used before calculators, by referring to numerical tables and performing manipulations such as those above.
High precision
To compute the natural logarithm with many digits of precision, the Taylor series approach is not efficient since the convergence is slow. An alternative is to use
Newton's method to invert the exponential function, whose series converges more quickly.
An alternative for extremely high precision calculation is the formula
where
M denotes the arithmetic-geometric mean and
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.
Complex logarithms
The exponential function can be extended to a function which gives a
complex number as the exponential of any arbitrary complex number; simply apply the infinite series with
x complex. This can be inverted to form a complex logarithm. This will obey much the same properties as the ordinary logarithm. There are two difficulties involved: no
x has e
x = 0; and it turns out that e
2pi = 1 = e
0. Since the multiplicative property still works, e
z = e
z+2pi, for all complex
z and integral
n.
So the logarithm cannot be defined for the whole
complex plane, and even then it is ambiguous; any logarithm can be changed by adding 2p
i at will. The result of this is that the complex logarithm can only be consistent within a limited area. Thus for example, log
i =1/2 p
i or 5/2p
i or -3/2 p
i etc.; and although
i4 = 1, 4 log
i will equal 2p
i, or 10p
i or -6p
i and so on.
See also
References
