Mathematicians, statisticians, and some engineers generally understand either "log(x)" or "ln(x)" to mean loge(x), i.e., the natural logarithm of x, and write "log10(x)" if the base 10 logarithm of x is intended.
Some engineers, biologists, and some others generally write "ln(x)" (or occasionally "loge(x)") when they mean the natural logarithm of x, and take "log(x)" to mean log10(x) or, in the case of some computer scientists, log2(x) (although this is often written lg(x) instead).
In hand-held calculators, the natural logarithm is denoted ln, whereas log is the base 10 logarithm.
In information theory and cryptography "log(x)" means "log2(x)".
Natural logarithm
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 2.718281828. It is also sometimes referred to as the Napierian logarithm, although the original meaning of this term is slightly different. 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.
Graph of the natural logarithm function. The function slowly grows to positive infinity as x increases and slowly goes to negative infinity as x approaches 0 ("slowly" as compared to any power law of x).
The natural logarithm function can also be defined as the inverse function of the exponential function, leading to the identities:
eln(x) = x , if x > 0
ln(ex) = 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. Represented as a function:
ln : R+à R
Logarithms can be defined to any positive base other than 1, not just e, and are useful for solving equations in which the unknown appears as the exponent of some other quantity.
Why it is called natural
Initially, it might seem that since our numbering system is base 10, this base would be more "natural" than base e. But mathematically, the number 10 is not particularly significant. Its use culturally—as the basis for many societies’ numbering systems—likely arises from humans’ typical number of fingers. Other cultures have based their counting systems on such choices as 5, 20, and 60.
loge is a “natural” log because it automatically springs from, and appears so often in, mathematics. For example, consider the problem of differentiating a logarithmic function:
If the base b equals e, then the derivative is simply 1/x, and at x = 1 this derivative equals 1. Another sense in which the base e logarithm is the most natural is that it can be defined quite easily in terms of a simple integral or Taylor series and this is not true of other logarithms.
Further senses of this naturalness make no use of calculus. As an example, there are a number of simple series involving the natural logarithm. In fact, Pietro Mengols and Nicholas Mercator called it logarithmus naturalis a few decades before Newton and Leibniz developed calculus
Definitions
ln(x) defined as the area under the curve f(x) = 1/x.
Formally, ln(a) may be defined as the area under the graph of 1/x from 1 to a, that is as the integral,
This defines a logarithm because it satisfies the fundamental property of a logarithm:
ln(ab) = ln(a) + ln(b)
This can be demonstrated by letting t = x/a as follows:
The number e can then be defined as the unique real number a such that ln(a) = 1.
Alternatively, if the exponential function has been defined first using an infinite series, the natural logarithm may be defined as its inverse function, i.e., ln(x) is that function such that eln(x) = x . Since the range of the exponential function on real arguments is all positive real numbers and since the exponential function is strictly increasing, this is well-defined for all positive x.
Derivative, Taylor series
The derivative of the natural logarithm is given by
The Taylor polynomials for loge(1+x) only provide accurate approximations in the range -1 < x ≤ 1. Note that, for x > 1, the Taylor polynomials of higher degree are worse approximations.
This leads to the Taylor series for ln(1 + x) around 0; also known as the Mercator series
unless x = -1
At right is a picture of ln(1 + x) and some of its Taylor polynomials around 0. These approximations converge to the function only in the region -1 < x ≤ 1; outside of this region the higher-degree Taylor polynomials are worse approximations for the function.
Substituting x-1 for x, we obtain an alternative form for ln(x) itself, namely
for |x - 1|<1
By using the Euler transform on the Mercator series, one obtains the following, which is valid for any x with absolute value greater than 1:
This series is similar to a BBP-type formula.
Expansions at a glance
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