Natural logarithm

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The natural logarithm of a number is its logarithm to the base of the mathematical constant e, where e is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, log_e x, or sometimes, if the base e is implicit, simply log x. Parentheses are sometimes added for clarity, giving ln(x), log_e(x) or log(x). This is done in particular when the argument to the logarithm is not a single symbol, to prevent ambiguity.

The natural logarithm of x is the power to which e would have to be raised to equal x. For example, ln(7.5) is 2.0149..., because e^2.0149... = 7.5. The natural log of e itself, ln(e), is 1, because e¹ = e, while the natural logarithm of 1, ln(1), is 0, since e⁰ = 1.

The natural logarithm can be defined for any positive real number a as the area under the curve y = 1/x from 1 to a (the area being taken as negative when a < 1). The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to the term "natural". The definition of the natural logarithm can be extended to give logarithm values for negative numbers and for all non-zero complex numbers, although this leads to a multi-valued function: see Complex logarithm.

The natural logarithm function, if considered as a real-valued function of a real variable, is the inverse function of the exponential function, leading to the identities:

${\displaystyle e^{\ln x}=x\qquad {\text{if }}x>0}$

${\displaystyle \ln(e^{x})=x.}$

Like all logarithms, the natural logarithm maps multiplication into addition:

${\displaystyle \ln(xy)=\ln x+\ln y.}$

Thus, the logarithm function is a group isomorphism from positive real numbers under multiplication to the group of real numbers under addition, represented as a function:

${\displaystyle \ln \colon \mathbb {R} ^{+}\to \mathbb {R} .}$

Logarithms can be defined to any positive base other than 1, not only e. However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, and are usually defined in terms of the latter. For instance, the binary logarithm is the natural logarithm divided by ln(2), the natural logarithm of 2. Logarithms are useful for solving equations in which the unknown appears as the exponent of some other quantity. For example, logarithms are used to solve for the half-life, decay constant, or unknown time in exponential decay problems. They are important in many branches of mathematics and the sciences and are used in finance to solve problems involving compound interest.

By Lindemann-Weierstrass theorem, the natural logarithm of any positive algebraic number other than 1 is a transcendental number.

Video Natural logarithm

History

The concept of the natural logarithm was worked out by Gregoire de Saint-Vincent and Alphonse Antonio de Sarasa before 1649. Their work involved quadrature of the hyperbola xy = 1 by determination of the area of hyperbolic sectors. Their solution generated the requisite "hyperbolic logarithm" function having properties now associated with the natural logarithm.

An early mention of the natural logarithm was by Nicholas Mercator in his work Logarithmotechnia published in 1668, although the mathematics teacher John Speidell had already in 1619 compiled a table of what in fact were effectively natural logarithms.

Maps Natural logarithm

Notational conventions

The notations "ln x" and "log_e x" both refer unambiguously to the natural logarithm of x. "log x" without an explicit base may also refer to the natural logarithm. This usage is common in mathematics and some scientific contexts as well as in many programming languages. In some other contexts, however, "log x" can be used to denote the common (base 10) logarithm.

Historically, the notations "l." and "l" were in use at least since the 1730s, and until at least the 1840s, then "log." or "log", at least since the 1790s. Finally, in the twentieth century, the notations "Log" and "logh" are attested.

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Origin of the term natural logarithm

The function ${\displaystyle n\mapsto (e^{n},e^{-n})}$ for n ? Z produces a bi-infinite sequence of hyperbolic points. When two adjacent points are joined to (0, 0) by hyperbolic radii, the hyperbolic sector so formed has unit area. Thus the total area inside the hyperbola and its asymptotes is infinite, consistent with divergence of the harmonic series. Area measure accords with the arc measure in both the circle and right hyperbola: in a circle of radius ?2, the arc of a circular sector has an angle equal to the sector area. Likewise, the hyperbolic angle of a hyperbolic arc is measured by the area of the corresponding hyperbolic sector of xy = 1.

Tribute is paid to Leonhard Euler who profiled the importance of Euler's number e = 2.71828... as the base of the exponential function and natural logarithm. He introduced the idea of a transcendental function to classify the trigonometric and exponential functions in the precalculus textbook Introduction to the Analysis of the Infinite (1748). The quadrature of the hyperbola requires the natural logarithm, so integral calculus was inhibited by the lack of an expression for hyperbolic quadrature until Gregoire de Saint-Vincent (1647) described it with a logarithmic feature: the correspondence of an arithmetic sequence of areas with a geometric sequence on the asymptote. Expositions by Nicholas Mercator, Christiaan Huygens and others led to Euler's Introduction that detailed the circular functions in terms of infinite series.

The connection between area and the arcs of circular and hyperbolic functions demonstrates the naturalness of this logarithm.

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Definitions

Formally, ln(a) may be defined as the area under the hyperbola 1/x. This is the integral,

${\displaystyle \ln a=\int _{1}^{a}{\frac {1}{x}}\,dx.}$

This function is a logarithm because it satisfies the fundamental property of a logarithm:

${\displaystyle \ln(ab)=\ln a+\ln b.}$

This can be demonstrated by splitting the integral that defines ln(ab) into two parts and then making the variable substitution x = ta in the second part, as follows:

${\displaystyle {\begin{aligned}\ln(ab)=\int _{1}^{ab}{\frac {1}{x}}\,dx&=\int _{1}^{a}{\frac {1}{x}}\,dx+\int _{a}^{ab}{\frac {1}{x}}\,dx\\[5pt]&=\int _{1}^{a}{\frac {1}{x}}\,dx+\int _{1}^{b}{\frac {1}{at}}\,d(at)\\[5pt]&=\int _{1}^{a}{\frac {1}{x}}\,dx+\int _{1}^{b}{\frac {1}{t}}\,dt\\[5pt]&=\ln a+\ln b.\end{aligned}}}$

In elementary terms, this is simply scaling by 1/a in the horizontal direction and by a in the vertical direction. Area does not change under this transformation, but the region between a and ab is reconfigured. Because the function a/(ax) is equal to the function 1/x, the resulting area is precisely ln(b).

The number e is defined as the unique real number a such that ln(a) = 1.

Alternatively, if the exponential function has been defined first, say by using an infinite series, the natural logarithm may be defined as its inverse function, i.e., ln is that function such that exp(ln(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.

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Properties

• ${\displaystyle \ln 1=0}$
• ${\displaystyle \ln e=1}$
• ${\displaystyle \ln(xy)=\ln x+\ln y,\quad {\text{for}}\quad x>0,y>0}$
• ${\displaystyle \ln(x^{y})=y\ln x\quad {\text{for}}\quad x>0}$
• ${\displaystyle \ln x<\ln y\quad {\text{for}}\quad 0<x<y}$
• ${\displaystyle \lim _{x\to 0}{\frac {\ln(1+x)}{x}}=1}$
• ${\displaystyle \lim _{n\to 0}{\frac {x^{n}-1}{n}}=\ln x}$
• ${\displaystyle {\frac {x-1}{x}}\leq \ln x\leq x-1\quad {\text{for}}\quad x>0}$
• ${\displaystyle \ln {(1+x^{\alpha })}\leq \alpha x\quad {\text{for}}\quad x\geq 0,\alpha \geq 1}$

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Derivative

The derivative of the natural logarithm as a real-valued function on the positive reals is given by

${\displaystyle {\frac {d}{dx}}\ln x={\frac {1}{x}}.}$

How to establish this derivative of the natural logarithm depends on how it is defined firsthand. If the natural logarithm is defined as the integral

${\displaystyle \ln x=\int _{1}^{x}{\frac {1}{t}}\,dt,}$

then the derivative immediately follows from the first part of the fundamental theorem of calculus.

If the natural logarithm is defined as the inverse of the (natural) exponential function, then the derivative for x > 0 can be found by using the properties of the logarithm and a definition of the exponential function.

${\displaystyle {\begin{aligned}{\frac {d}{dx}}\ln x&=\lim _{h\to 0}{\frac {\ln(x+h)-\ln x}{h}}\\&=\lim _{h\to 0}\left({\frac {1}{h}}\ln \left({\frac {x+h}{x}}\right)\right)\\&=\lim _{h\to 0}\ln \left(1+{\frac {h}{x}}\right)^{\frac {1}{h}}\quad &&{\text{all above for logarithmic properties}}\\&=\ln \lim _{h\to 0}\left(1+{\frac {h}{x}}\right)^{\frac {1}{h}}\quad &&{\text{for continuity of the logarithm}}\\&=\ln e^{1/x}\quad &&{\text{for the definition of }}e^{x}=\lim _{h\to 0}(1+hx)^{(1/h)}\\&={\frac {1}{x}}\quad &&{\text{for the definition of the ln as inverse function.}}\end{aligned}}}$

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Series

If ${\displaystyle \textstyle \vert x-1\vert \leq 1{\text{ and }}x\neq 0}$, then

${\displaystyle {\begin{aligned}\ln(x)&=\int _{1}^{x}{\frac {1}{t}}\,dt=\int _{0}^{x-1}{\frac {1}{1+u}}\,du\\&=\int _{0}^{x-1}(1-u+u^{2}-u^{3}+\cdots )\,du\\&=(x-1)-{\frac {(x-1)^{2}}{2}}+{\frac {(x-1)^{3}}{3}}-{\frac {(x-1)^{4}}{4}}+\cdots \\&=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}(x-1)^{k}}{k}}.\end{aligned}}}$

This is also the Taylor series for ln(x) around 1. Alternatively, it can be written as

${\displaystyle \ln(1+x)=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k}}x^{k}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-\cdots }$

Leonhard Euler, disregarding ${\displaystyle x\neq 0}$, nevertheless applied this series to x = -1, in order to show that the harmonic series equals the (natural) logarithm of 1/(1 - 1), that is the logarithm of infinity. Nowadays, more formally, one can prove that the harmonic series truncated at N is close to the logarithm of N, when N is large.

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 evolve to worse approximations for the function.

A useful special case for positive integers n, taking ${\displaystyle x={\tfrac {1}{n}}}$, is:

${\displaystyle \ln \left({\frac {n+1}{n}}\right)=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{kn^{k}}}={\frac {1}{n}}-{\frac {1}{2n^{2}}}+{\frac {1}{3n^{3}}}-{\frac {1}{4n^{4}}}+\cdots }$

If ${\displaystyle \operatorname {Re} (x)\geq 1/2,}$ then

${\displaystyle {\begin{aligned}\ln(x)&=-\ln \left({\frac {1}{x}}\right)=-\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}({\frac {1}{x}}-1)^{k}}{k}}=\sum _{k=1}^{\infty }{\frac {(x-1)^{k}}{kx^{k}}}\\&={\frac {x-1}{x}}+{\frac {(x-1)^{2}}{2x^{2}}}+{\frac {(x-1)^{3}}{3x^{3}}}+{\frac {(x-1)^{4}}{4x^{4}}}+\cdots \end{aligned}}}$

Now taking ${\displaystyle x={\tfrac {n+1}{n}}}$ for positive integers n, yields:

${\displaystyle \ln \left({\frac {n+1}{n}}\right)=\sum _{k=1}^{\infty }{\frac {1}{k(n+1)^{k}}}={\frac {1}{n+1}}+{\frac {1}{2(n+1)^{2}}}+{\frac {1}{3(n+1)^{3}}}+{\frac {1}{4(n+1)^{4}}}+\cdots }$

If ${\displaystyle \operatorname {Re} (x)\geq 0{\text{ and }}x\neq 0,}$ then

${\displaystyle \ln(x)=\ln \left({\frac {2x}{2}}\right)=\ln \left({\frac {1+{\frac {x-1}{x+1}}}{1-{\frac {x-1}{x+1}}}}\right)=\ln \left(1+{\frac {x-1}{x+1}}\right)-\ln \left(1-{\frac {x-1}{x+1}}\right).}$

Since

${\displaystyle {\begin{aligned}\ln(1+y)-\ln(1-y)&=\sum _{i=1}^{\infty }{\frac {1}{i}}\left((-1)^{i-1}y^{i}-(-1)^{i-1}(-y)^{i}\right)=\sum _{i=1}^{\infty }{\frac {y^{i}}{i}}\left((-1)^{i-1}+1\right)\\&=y\sum _{i=1}^{\infty }{\frac {y^{i-1}}{i}}\left((-1)^{i-1}+1\right){\overset {i-1\to 2k}{=}}\;2y\sum _{k=0}^{\infty }{\frac {y^{2k}}{2k+1}},\end{aligned}}}$

we arrive at

${\displaystyle {\begin{aligned}\ln(x)&={\frac {2(x-1)}{x+1}}\sum _{k=0}^{\infty }{\frac {1}{2k+1}}{\left({\frac {(x-1)^{2}}{(x+1)^{2}}}\right)}^{k}\\&={\frac {2(x-1)}{x+1}}\left({\frac {1}{1}}+{\frac {1}{3}}{\frac {(x-1)^{2}}{(x+1)^{2}}}+{\frac {1}{5}}{\left({\frac {(x-1)^{2}}{(x+1)^{2}}}\right)}^{2}+\cdots \right).\end{aligned}}}$

Substituting again ${\displaystyle x={\tfrac {n+1}{n}}}$ for positive integers n, yields:

${\displaystyle {\begin{aligned}\ln \left({\frac {n+1}{n}}\right)&={\frac {2}{2n+1}}\sum _{k=0}^{\infty }{\frac {1}{(2k+1)((2n+1)^{2})^{k}}}\\&=2\left({\frac {1}{2n+1}}+{\frac {1}{3(2n+1)^{3}}}+{\frac {1}{5(2n+1)^{5}}}+\cdots \right).\end{aligned}}}$

This is, by far, the fastest converging of the series described here.

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The natural logarithm in integration

The natural logarithm allows simple integration of functions of the form g(x) = f '(x)/f(x): an antiderivative of g(x) is given by ln(|f(x)|). This is the case because of the chain rule and the following fact:

${\displaystyle {\frac {d}{dx}}\ln \left|x\right|={\frac {1}{x}}.}$

In other words,

${\displaystyle \int {\frac {1}{x}}\,dx=\ln |x|+C}$

and

${\displaystyle \int {{\frac {f'(x)}{f(x)}}\,dx}=\ln |f(x)|+C.}$

Here is an example in the case of g(x) = tan(x):

${\displaystyle {\begin{aligned}&\int \tan x\,dx=\int {\frac {\sin x}{\cos x}}\,dx\\[6pt]&\int \tan x\,dx=\int {\frac {-{\frac {d}{dx}}\cos x}{\cos x}}\,dx.\end{aligned}}}$

Letting f(x) = cos(x):

${\displaystyle \int \tan x\,dx=-\ln \left|\cos x\right|+C}$

${\displaystyle \int \tan x\,dx=\ln \left|\sec x\right|+C}$

where C is an arbitrary constant of integration.

The natural logarithm can be integrated using integration by parts:

${\displaystyle \int \ln x\,dx=x\ln x-x+C.}$

Let:

${\displaystyle u=\ln x\Rightarrow du={\frac {dx}{x}}}$

${\displaystyle dv=dx\Rightarrow v=x}$

then:

${\displaystyle {\begin{aligned}\int \ln x\,dx&=x\ln x-\int {\frac {x}{x}}\,dx\\&=x\ln x-\int 1\,dx\\&=x\ln x-x+C\end{aligned}}}$

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Numerical value

For ln(x) 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:

${\displaystyle {\begin{aligned}\ln 123.456&=\ln(1.23456\cdot 10^{2})\\&=\ln 1.23456+\ln(10^{2})\\&=\ln 1.23456+2\ln 10\\&\approx \ln 1.23456+2\cdot 2.3025851.\end{aligned}}}$

Such techniques were used before calculators, by referring to numerical tables and performing manipulations such as those above.

Natural logarithm of 10

The natural logarithm of 10, which has the decimal expansion 2.30258509..., plays a role for example in the computation of natural logarithms of numbers represented in scientific notation, as a mantissa multiplied by a power of 10:

${\displaystyle \ln(a\cdot 10^{n})=\ln a+n\ln 10.}$

This means that one can effectively calculate the logarithms of numbers with very large or very small magnitude using the logarithms of a relatively small set of decimals in the range ${\displaystyle [1,10)}$.

High precision

To compute the natural logarithm with many digits of precision, the Taylor series approach is not efficient since the convergence is slow. Especially if x is near 1, a good alternative is to use Halley's method or Newton's method to invert the exponential function, because the series of the exponential function converges more quickly. For finding the value of y to give exp(y) - x = 0 using Halley's method, or equivalently to give exp(y/2) - x exp(-y/2) = 0 using Newton's method, the iteration simplifies to

${\displaystyle y_{n+1}=y_{n}+2\cdot {\frac {x-\exp(y_{n})}{x+\exp(y_{n})}}}$

which has cubic convergence to ln(x).

Another alternative for extremely high precision calculation is the formula

${\displaystyle \ln x\approx {\frac {\pi }{2M(1,4/s)}}-m\ln 2,}$

where M denotes the arithmetic-geometric mean of 1 and 4/s, and

${\displaystyle s=x2^{m}>2^{p/2},}$

with m chosen so that p bits of precision is attained. (For most purposes, the value of 8 for m is sufficient.) 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 ? can be pre-computed to the desired precision using any of several known quickly converging series.)

Based on a proposal by William Kahan and first implemented in the Hewlett-Packard HP-41C calculator in 1979 (referred to under "LN1" in the display, only), some calculators, computer algebra systems and programming languages (for example C99) provide a special natural logarithm plus 1 function, alternatively named LNP1, or log1p to give more accurate results for logarithms close to zero by passing arguments x, also close to zero, to a function log1p(x), which returns the value ln(1+x), instead of passing a value y close to 1 to a function returning ln(y). The function log1p avoids in the floating point arithmetic a near cancelling of the absolute term 1 with the second term from the Taylor expansion of the ln, thereby allowing for a high accuracy for both the argument and the result near zero. Similar inverse functions named "expm1", "expm" or "exp1m" exist as well, all with the meaning of expm1(x) = exp(x) - 1.

An identity in terms of the inverse hyperbolic tangent,

${\displaystyle \mathrm {log1p} (x)=\log(1+x)=2~\mathrm {artanh} \left({\frac {x}{2+x}}\right)\,,}$

gives a high precision value for small values of x on systems that do not implement log1p(x).

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.

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Continued fractions

While no simple continued fractions are available, several generalized continued fractions are, including:

${\displaystyle {\begin{aligned}\ln(1+x)&={\frac {x^{1}}{1}}-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+{\frac {x^{5}}{5}}-\cdots \\[5pt]&={\cfrac {x}{1-0x+{\cfrac {1^{2}x}{2-1x+{\cfrac {2^{2}x}{3-2x+{\cfrac {3^{2}x}{4-3x+{\cfrac {4^{2}x}{5-4x+\ddots }}}}}}}}}}\end{aligned}}}$

${\displaystyle {\begin{aligned}\ln \left(1+{\frac {x}{y}}\right)&={\cfrac {x}{y+{\cfrac {1x}{2+{\cfrac {1x}{3y+{\cfrac {2x}{2+{\cfrac {2x}{5y+{\cfrac {3x}{2+\ddots }}}}}}}}}}}}\\[5pt]&={\cfrac {2x}{2y+x-{\cfrac {(1x)^{2}}{3(2y+x)-{\cfrac {(2x)^{2}}{5(2y+x)-{\cfrac {(3x)^{2}}{7(2y+x)-\ddots }}}}}}}}\end{aligned}}}$

These continued fractions--particularly the last--converge rapidly for values close to 1. However, the natural logarithms of much larger numbers can easily be computed by repeatedly adding those of smaller numbers, with similarly rapid convergence.

For example, since 2 = 1.25³ × 1.024, the natural logarithm of 2 can be computed as:

${\displaystyle {\begin{aligned}\ln 2&=3\ln \left(1+{\frac {1}{4}}\right)+\ln \left(1+{\frac {3}{125}}\right)\\[8pt]&={\cfrac {6}{9-{\cfrac {1^{2}}{27-{\cfrac {2^{2}}{45-{\cfrac {3^{2}}{63-\ddots }}}}}}}}+{\cfrac {6}{253-{\cfrac {3^{2}}{759-{\cfrac {6^{2}}{1265-{\cfrac {9^{2}}{1771-\ddots }}}}}}}}.\end{aligned}}}$

Furthermore, since 10 = 1.25¹⁰ × 1.024³, even the natural logarithm of 10 similarly can be computed as:

${\displaystyle {\begin{aligned}\ln 10&=10\ln \left(1+{\frac {1}{4}}\right)+3\ln \left(1+{\frac {3}{125}}\right)\\[10pt]&={\cfrac {20}{9-{\cfrac {1^{2}}{27-{\cfrac {2^{2}}{45-{\cfrac {3^{2}}{63-\ddots }}}}}}}}+{\cfrac {18}{253-{\cfrac {3^{2}}{759-{\cfrac {6^{2}}{1265-{\cfrac {9^{2}}{1771-\ddots }}}}}}}}.\end{aligned}}}$

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Complex logarithms

The exponential function can be extended to a function which gives a complex number as e^x 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 e^x = 0; and it turns out that e^2?i = 1 = e⁰. Since the multiplicative property still works for the complex exponential function, e^z = e^z+2?ki, for all complex z and integers k.

So the logarithm cannot be defined for the whole complex plane, and even then it is multi-valued - any complex logarithm can be changed into an "equivalent" logarithm by adding any integer multiple of 2?i at will. The complex logarithm can only be single-valued on the cut plane. For example, ln(i) = ?i/2 or 5?i/2 or -3?i/2, etc.; and although i⁴ = 1, 4 log(i) can be defined as 2?i, or 10?i or -6?i, and so on.

• Plots of the natural logarithm function on the complex plane (principal branch)

See also

• Approximating natural exponents (log base e)
• John Napier - discoverer of logarithms
• Logarithm of a matrix
• Logarithmic integral function
• Nicholas Mercator - first to use the term natural logarithm
• Polylogarithm
• Von Mangoldt function
• The number e

Notes

References

Source of article : Wikipedia