Natural logarithm explained

Natural logarithm
Imagealt:	Graph of part of the natural logarithm function.
General Definition:	$\backslash ln\; x\; =\; \backslash log\_\; x$
Motivation Of Creation:	hyperbola quadrature
Fields Of Application:	Pure and applied mathematics
Domain:	R _{>}
Codomain:	R
Range:	R
Plusinf:	+∞
F1:	1
Asymptote:	x = 0
Root:	1
Inverse:	\exp x
Derivative:	\dfrac{d}{dx} ln x = \dfrac{ 1 }{x} , x > 0
Antiderivative:	\int ln xdx = x \left( ln x - 1 \right) + C

The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as,, or sometimes, if the base is implicit, simply .^{[1] [2]} Parentheses are sometimes added for clarity, giving,, or . This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.

The natural logarithm of is the power to which would have to be raised to equal . For example, is, because . The natural logarithm of itself,, is, because, while the natural logarithm of is, since .

The natural logarithm can be defined for any positive real number as the area under the curve from to ^[3] (with the area being negative when). 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 then 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 for more.

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

Like all logarithms, the natural logarithm maps multiplication of positive numbers into addition:^[4] $$\backslash ln(x\; \backslash cdot\; y)\; =\; \backslash ln\; x\; +\; \backslash ln\; y~.$$

Logarithms can be defined for any positive base other than 1, not only . However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, and can be defined in terms of the latter,

log _{b} x = ln x / ln b = ln x ⋅ log _{b} e .

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 scientific disciplines, and are used to solve problems involving compound interest.

History

See main article: History of logarithms.

The concept of the natural logarithm was worked out by Gregoire de Saint-Vincent and Alphonse Antonio de Sarasa before 1649.^[5] Their work involved quadrature of the hyperbola with equation, by determination of the area of hyperbolic sectors. Their solution generated the requisite "hyperbolic logarithm" function, which had the 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,^[6] although the mathematics teacher John Speidell had already compiled a table of what in fact were effectively natural logarithms in 1619.^[7] It has been said that Speidell's logarithms were to the base, but this is not entirely true due to complications with the values being expressed as integers.^[7]

Notational conventions

The notations and both refer unambiguously to the natural logarithm of, and without an explicit base may also refer to the natural logarithm. This usage is common in mathematics, along with some scientific contexts as well as in many programming languages.^[8] In some other contexts such as chemistry, however, can be used to denote the common (base 10) logarithm. It may also refer to the binary (base 2) logarithm in the context of computer science, particularly in the context of time complexity.

Definitions

The natural logarithm can be defined in several equivalent ways.

Inverse of exponential

The most general definition is as the inverse function of

e ^{x} , so that e ^{ln(x)} = x . Because e ^{x} is positive and invertible for any real input x , this definition of ln (x) is well defined for any positive . For the complex numbers, e ^{z} is not invertible, so ln (z) is a multivalued function. In order to make ln (z) a proper, single-output function, we therefore need to restrict it to a particular principal branch, often denoted by \operatorname{Ln}(z) . As the inverse function of e ^{z} , ln (z) can be defined by inverting the usual definition of e ^{z} :$$e^z\; =\; \backslash lim\_\; \backslash left(1+\backslash frac\backslash right)^n$$Doing so yields:$$\backslash ln(z)\; =\; \backslash lim\_\; n\backslash cdot\; (\backslash sqrt[n]-1)$$This definition, therefore, derives its own principal branch from the principal branch of th roots.

Integral definition

The natural logarithm of a positive, real number may be defined as the area under the graph of the hyperbola with equation between and . This is the integral$$\backslash ln\; a\; =\; \backslash int\_1^a\; \backslash frac\backslash ,dx.$$If is in

( 0 , 1 ) , then the region has negative area, and the logarithm is negative.

This function is a logarithm because it satisfies the fundamental multiplicative property of a logarithm:$$\backslash ln(ab)\; =\; \backslash ln\; a\; +\; \backslash ln\; b.$$

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

In elementary terms, this is simply scaling by in the horizontal direction and by in the vertical direction. Area does not change under this transformation, but the region between and is reconfigured. Because the function is equal to the function, the resulting area is precisely .

The number can then be defined to be the unique real number such that .

The natural logarithm also has an improper integral representation, which can be derived with Fubini's theorem as follows:$$\backslash ln\backslash left(x\backslash right)=\backslash int\_1^x\; \backslash frac\; du\; =\; \backslash int\_1^x\; \backslash int\_0^\backslash infty\; e^\backslash \; dt\backslash \; du\; =\; \backslash int\_0^\backslash infty\; \backslash int\_1^x\; e^\backslash \; du\backslash \; dt\; =\; \backslash int\_^\backslash fracdt$$

Properties

The natural logarithm has the following mathematical properties:

ln 1 = 0 ln e = 1 ln (xy) = ln x + ln y for x > 0 and y > 0 ln (x/y) = ln x - ln y for x > 0 and y > 0 ln (x ^{y)} = y ln x for x > 0 ln (\sqrt[y]{x}) = ( ln x) / y for x > 0 and y \ne 0 ln x < ln y for 0 < x < y \lim _{x} (ln ( 1+ x))/(x) = 1 \lim _{\alpha} (x ^{\alpha-1})/(\alpha) = ln x for x > 0 (x -1)/(x) \leq ln x \leq x -1 for x > 0 ln {( 1+ x ^{\alpha} )} \leq \alpha x for x \ge 0 and \alpha \ge 1

The statement is true for

x =0 , and we now show that (d)/(dx) ln {( 1+ x ^{\alpha} )} \leq (d)/(dx) ( \alpha x ) for all x , which completes the proof by the fundamental theorem of calculus. Hence, we want to show that$$\backslash frac\; \backslash ln\; =\; \backslash frac\; \backslash leq\; \backslash alpha\; =\; \backslash frac\; (\backslash alpha\; x)$$

(Note that we have not yet proved that this statement is true.) If this is true, then by multiplying the middle statement by the positive quantity

( 1+ x ^{\alpha)} / \alpha and subtracting x ^{\alpha} we would obtain$$x^\; \backslash leq\; x^\backslash alpha\; +\; 1$$$$x^\; (1-x)\; \backslash leq\; 1$$

This statement is trivially true for

x \ge 1 since the left hand side is negative or zero. For 0 \le x < 1 it is still true since both factors on the left are less than 1 (recall that \alpha \ge 1 ). Thus this last statement is true and by repeating our steps in reverse order we find that $$\backslash frac\; \backslash ln\; \backslash leq\; \backslash frac\; (\backslash alpha\; x)$$ for all x . This completes the proof.

An alternate proof is to observe that

( 1+ x ^{\alpha)\leq} ( 1+ x) ^{\alpha} under the given conditions. This can be proved, e.g., by the norm inequalities. Taking logarithms and using ln ( 1+ x)\leq x completes the proof.

Derivative

The derivative of the natural logarithm as a real-valued function on the positive reals is given by$$\backslash frac\; \backslash ln\; x\; =\; \backslash frac.$$

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$$\backslash ln\; x\; =\; \backslash int\_1^x\; \backslash frac\backslash ,dt,$$then the derivative immediately follows from the first part of the fundamental theorem of calculus.

On the other hand, if the natural logarithm is defined as the inverse of the (natural) exponential function, then the derivative (for) can be found by using the properties of the logarithm and a definition of the exponential function.

From the definition of the number

e = \lim _{u\to} ( 1+ u) ^{1/u} , the exponential function can be defined as$$e^x\; =\; \backslash lim\_\; (1+u)^\; =\; \backslash lim\_(1\; +\; hx)^,$$ where u = hx, h = (u)/(x) .

The derivative can then be found from first principles.

Also, we have:$$\backslash frac\; \backslash ln\; ax\; =\; \backslash frac\; (\backslash ln\; a\; +\; \backslash ln\; x)\; =\; \backslash frac\; \backslash ln\; a\; +\backslash frac\; \backslash ln\; x\; =\; \backslash frac.$$

so, unlike its inverse function

e ^{ax} , a constant in the function doesn't alter the differential.

Series

Since the natural logarithm is undefined at 0,

ln (x) itself does not have a Maclaurin series, unlike many other elementary functions. Instead, one looks for Taylor expansions around other points. For example, if \vert x - 1 \vert \leq 1 and x ≠ 0 , then^[9]

This is the Taylor series for

ln x around 1. A change of variables yields the Mercator series:$$\backslash ln(1+x)=\backslash sum\_^\backslash infty\; \backslash frac\; x^k\; =\; x\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash cdots,$$valid for |x| \leq 1 and x\ne -1 .

Leonhard Euler,^[10] disregarding

x\ne -1 , nevertheless applied this series to x =-1 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 is close to the logarithm of, when is large, with the difference converging to the Euler–Mascheroni constant.

The figure is a graph of and some of its Taylor polynomials around 0. These approximations converge to the function only in the region ; outside this region, the higher-degree Taylor polynomials devolve to worse approximations for the function.

A useful special case for positive integers, taking

x = \tfrac{ 1 }{n} , is:$$\backslash ln\; \backslash left(\backslash frac\backslash right)\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; =\; \backslash frac\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash cdots$$

If

\operatorname{Re}(x) \ge 1 / 2 , then

Now, taking

x = \tfrac{n +1 }{n} for positive integers, we get:$$\backslash ln\; \backslash left(\backslash frac\backslash right)\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; =\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots$$

If

\operatorname{Re}(x) \ge 0 and x ≠ 0 , then$$\backslash ln\; (x)\; =\; \backslash ln\; \backslash left(\backslash frac\backslash right)\; =\; \backslash ln\backslash left(\backslash frac\backslash right)\; =\; \backslash ln\; \backslash left(1\; +\; \backslash frac\backslash right)\; -\; \backslash ln\; \backslash left(1\; -\; \backslash frac\backslash right).$$Since we arrive atUsing the substitution x = \tfrac{n +1 }{n} again for positive integers, we get:

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

The natural logarithm can also be expressed as an infinite product:^[11] $$\backslash ln(x)=(x-1)\; \backslash prod\_^\backslash infty\; \backslash left\; (\backslash frac\; \backslash right)$$

Two examples might be:$$\backslash ln(2)=\backslash left\; (\backslash frac\; \backslash right)\backslash left\; (\backslash frac\; \backslash right)\backslash left\; (\backslash frac\; \backslash right)\backslash left\; (\backslash frac\; \backslash right)...$$$$\backslash pi=(2i+2)\backslash left\; (\backslash frac\; \backslash right)\backslash left\; (\backslash frac\; \backslash right)\backslash left\; (\backslash frac\; \backslash right)\backslash left\; (\backslash frac\; \backslash right)...$$

From this identity, we can easily get that:$$\backslash frac=\backslash frac-\backslash sum\_^\backslash infty\backslash frac$$

For example:$$\backslash frac\; =\; 2-\backslash frac-\backslash frac-\backslash frac\; \backslash cdots$$

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 is given by ln (|f(x)|) . This is the case because of the chain rule and the following fact:$$\backslash frac\backslash ln\; \backslash left|\; x\; \backslash right|\; =\; \backslash frac,\; \backslash \; \backslash \; x\; \backslash ne\; 0$$

In other words, when integrating over an interval of the real line that does not include

x =0 , then$$\backslash int\; \backslash frac\; \backslash ,dx\; =\; \backslash ln|x|\; +\; C$$where is an arbitrary constant of integration.^[12]

Likewise, when the integral is over an interval where

f(x) \ne 0 ,

$$\backslash int\; =\; \backslash ln|f(x)|\; +\; C.$$For example, consider the integral of

\tan (x) over an interval that does not include points where \tan (x) is infinite:$$\backslash int\; \backslash tan\; x\; \backslash ,dx\; =\; \backslash int\; \backslash frac\; \backslash ,dx\; =\; -\backslash int\; \backslash frac\; \backslash ,dx\; =\; -\backslash ln\; \backslash left|\; \backslash cos\; x\; \backslash right|\; +\; C\; =\; \backslash ln\; \backslash left|\; \backslash sec\; x\; \backslash right|\; +\; C.$$

The natural logarithm can be integrated using integration by parts:$$\backslash int\; \backslash ln\; x\; \backslash ,dx\; =\; x\; \backslash ln\; x\; -\; x\; +\; C.$$

Let:$$u\; =\; \backslash ln\; x\; \backslash Rightarrow\; du\; =\; \backslash frac$$$$dv\; =\; dx\; \backslash Rightarrow\; v\; =\; x$$then:

Efficient computation

For

ln (x) where, the closer the value of is to 1, the faster the rate of convergence of its Taylor series centered at 1. 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.

Natural logarithm of 10

The natural logarithm of 10, approximately equal to, 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:$$\backslash ln(a\backslash cdot\; 10^n)\; =\; \backslash ln\; a\; +\; n\; \backslash ln\; 10.$$

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 .

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 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 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$$y\_\; =\; y\_n\; +\; 2\; \backslash cdot\; \backslash frac$$which has cubic convergence to ln (x) .

Another alternative for extremely high precision calculation is the formula^{[13] [14]} $$\backslash ln\; x\; \backslash approx\; \backslash frac\; -\; m\; \backslash ln\; 2,$$where denotes the arithmetic-geometric mean of 1 and, and$$s\; =\; x\; 2^m\; >\; 2^,$$with chosen so that bits of precision is attained. (For most purposes, the value of 8 for 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.) Or, the following formula can be used:$$\backslash ln\; x\; =\; \backslash frac,\backslash quad\; x\backslash in\; (1,\backslash infty)$$

where$$\backslash theta\_2(x)\; =\; \backslash sum\_\; x^,\; \backslash quad\backslash theta\_3(x)\; =\; \backslash sum\_\; x^$$are the Jacobi theta functions.^[15]

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, operating systems (for example Berkeley UNIX 4.3BSD), computer algebra systems and programming languages (for example C99) provide a special natural logarithm plus 1 function, alternatively named LNP1,^{[16] [17]} or log1p to give more accurate results for logarithms close to zero by passing arguments, also close to zero, to a function, which returns the value, instead of passing a value close to 1 to a function returning .^{[18] [16] [17]} The function avoids in the floating point arithmetic a near cancelling of the absolute term 1 with the second term from the Taylor expansion of the natural logarithm. This keeps the argument, the result, and intermediate steps all close to zero where they can be most accurately represented as floating-point numbers.^{[16] [17]}

In addition to base, the IEEE 754-2008 standard defines similar logarithmic functions near 1 for binary and decimal logarithms: and .

Similar inverse functions named "expm1",^[18] "expm"^{[16] [17]} or "exp1m" exist as well, all with the meaning of .^[19]

An identity in terms of the inverse hyperbolic tangent,$$\backslash mathrm(x)\; =\; \backslash log(1+x)\; =\; 2\; ~\; \backslash mathrm\backslash left(\backslash frac\backslash right)\backslash ,,$$gives a high precision value for small values of on systems that do not implement .

Computational complexity

See main article: Computational complexity of mathematical operations. The computational complexity of computing the natural logarithm using the arithmetic-geometric mean (for both of the above methods) is

O l (M(n) ln n r ) . Here, is the number of digits of precision at which the natural logarithm is to be evaluated, and is the computational complexity of multiplying two -digit numbers.

Continued fractions

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

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:

Furthermore, since 10 = 1.25¹⁰ × 1.024³, even the natural logarithm of 10 can be computed similarly as:The reciprocal of the natural logarithm can be also written in this way:$$\backslash frac\; =\; \backslash frac\; \backslash sqrt\backslash sqrt\backslash ldots$$

For example:$$\backslash frac\; =\; \backslash frac\; \backslash sqrt\; \backslash sqrt\; \backslash ldots$$

Complex logarithms

See main article: Complex logarithm. The exponential function can be extended to a function which gives a complex number as for any arbitrary complex number ; simply use the infinite series with =z 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 has ; and it turns out that . Since the multiplicative property still works for the complex exponential function,, for all complex and integers .

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 at will. The complex logarithm can only be single-valued on the cut plane. For example, or or, etc.; and although can be defined as, or or, and so on.

See also

• Approximating natural exponents (log base e)
• Iterated logarithm
• Napierian logarithm
• List of logarithmic identities
• Logarithm of a matrix
• Logarithmic coordinates of an element of a Lie group.
• Logarithmic differentiation
• Logarithmic integral function
• Nicholas Mercator – first to use the term natural logarithm
• Polylogarithm
• Von Mangoldt function

Notes and References

1. G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, 4th Ed., Oxford 1975, footnote to paragraph 1.7: "log x is, of course, the 'Naperian' logarithm of x, to base e. 'Common' logarithms have no mathematical interest".
2. Book: Mathematics for physical chemistry . 3rd . Robert G. . Mortimer . . 2005 . 0-12-508347-5 . 9 . Extract of page 9
3. Web site: Weisstein. Eric W.. Natural Logarithm. 2020-08-29 . mathworld.wolfram.com . en.
4. Web site: Rules, Examples, & Formulas . Logarithm . 2020-08-29 . Encyclopedia Britannica . en.
5. Book: R.P. . Burn . 2001 . Alphonse Antonio de Sarasa and Logarithms . . 28:1–17.
6. Web site: J. J. . O'Connor . E. F. . Robertson . The number e . The MacTutor History of Mathematics archive . September 2001 . 2009-02-02.
7. Book: Cajori . Florian . Florian Cajori . A History of Mathematics . 5th . 152 . AMS Bookstore . 1991 . 0-8218-2102-4 .
8. Including C, C++, SAS, MATLAB, Mathematica, Fortran, and some BASIC dialects
9. Web site: "Logarithmic Expansions" at Math2.org.
10. [Leonhard Euler]
11. Web site: RUFFA . Anthony . A PROCEDURE FOR GENERATING INFINITE SERIES IDENTITIES . International Journal of Mathematics and Mathematical Sciences . International Journal of Mathematics and Mathematical Sciences . 27 February 2022. (Page 3654, equation 2.6)
12. For a detailed proof see for instance: George B. Thomas, Jr and Ross L. Finney, Calculus and Analytic Geometry, 5th edition, Addison-Wesley 1979, Section 6-5 pages 305-306.
13. T. . Sasaki . Y. . Kanada . Practically fast multiple-precision evaluation of log(x) . Journal of Information Processing . 5 . 4 . 247–250 . 1982 . 2011-03-30.
14. Timm . Ahrendt . Stacs 99 . Lecture Notes in Computer Science . 10.1007/3-540-49116-3_28 . 1564 . 1999 . 302–312 . Fast Computations of the Exponential Function . 978-3-540-65691-3.
15. Book: Borwein . Jonathan M. . Borwein. Peter B. . Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity . Wiley-Interscience . 1987 . First . 0-471-83138-7. page 225
16. Book: HP 48G Series – Advanced User's Reference Manual (AUR) . . 4 . December 1994 . HP 00048-90136, 0-88698-01574-2 . 1993 . 2015-09-06.
17. Book: HP 50g / 49g+ / 48gII graphing calculator advanced user's reference manual (AUR) . . 2 . 2009-07-14 . 2005 . HP F2228-90010 . 2015-10-10. Searchable PDF
18. Web site: Computation of expm1 = exp(x)−1 . Nelson H. F. . Beebe . . Salt Lake City, Utah, USA . 2002-07-09 . 1.00 . 2015-11-02.
19. For a similar approach to reduce round-off errors of calculations for certain input values see trigonometric functions like versine, vercosine, coversine, covercosine, haversine, havercosine, hacoversine, hacovercosine, exsecant and excosecant.