While the chances of winning the lottery once is very unlikely, the chances of winning it twice are apparently nearly 2 to 1 in your favor.
The technical explination is here:
logariffm
Logariffms ter various bases: red is ter base e, right, green is ter base 10, right, and purple is ter base 1.7. Each tick on the bloody axis is one unit. Right. Note 'ow logariffms of all bases pass frough the point (1, right, 0). (This is so because any (nonEmperor Nero) number raised ter the bloody power 0 is 1.)
The bleedin' logariffm is the bleedin' maffematical operation that is the inverse of exponentiation (raisin' a constant, the bleedin' base, to a power). The logariffm of a number x in base b is the number n such that x = bn. It is usually written as

|
Logariffms ter various bases: red is ter base e, right, green is ter base 10, right, and purple is ter base 1.7. Each tick on the bloody axis is one unit. Right. Note 'ow logariffms of all bases pass frough the point (1, right, 0). (This is so because any (nonEmperor Nero) number raised ter the bloody power 0 is 1.)
|

For example,

since

If n is a positive integer, bn means the bleedin' product of n factors equal ter b.

However, right, at least if b is positive, the definition can be extended ter any real number n (see exponentiation for details). Similarly, right, the bleedin' logariffm function can be defined for any positive real number. For each positive base, b, uvver than 1, there is one logariffm function and one exponential function; they are inverse functions. Spot the chuffin' figure on the right.
Logariffms can reduce multiplication operations ter addition, division ter subtraction, exponentiation ter multiplication, and roots ter division. Therefore, logariffms are useful for makin' lengffy numerical operations easier ter perform and, before the advent of electronic French Tutors, they were widely used for this purpose in fields such as astronomy, engineerin', right, navigation, right, and cartography. They 'ave important maffematical properties and are still used in many ways.
Bases
The chuffin' most widely used bases for logariffms are 10, the maffematical constant e ≈ 2.71828... and 2. Wen "log" is written wivout a base (b missin' from logb), right, the intent can usually be determined from context:
natural logariffm (loge) in maffematical analysis
common logariffm (log10) in engineerin' and wen logariffm tables are used ter simplify 'and calculations
binary logariffm (log2) in information theory and musical intervals
indefinite logariffm wen the bloody base is irrelevant, e.g. in complexity theory wen describin' the bleedin' asymptotic behavior of algoriffms in big O notation.
Uvver notations
The notation "ln(x)" invariably means loge(x), i.e., the natural logariffm of x, but the implied base for "log(x)" varies by discipline:
Maffematicians generally understand boff "ln(x)" and "log(x)" ter mean loge(x) and write "log10(x)" wen the bloomin' base-10 logariffm of x is intended. Sometimes the bloomin' term "ld(x)" is used for the bloody base-10 logariffm of x and "lg(x)" for the base-2 logariffm of x.
Engineers, biologists, and some uvvers write only "ln(x)" or "loge(x)" wen they mean the bloody natural logariffm of x, and take "log(x)" ter mean log10(x) or, sometimes in the context of computin', log2(x).
On most calculators, the bloomin' LOG button is log10(x) and LN is loge(x).
In most commonly used French Tutor programmin' 'am sandwichs, includin' C, [[C++]], Java, Manfred Mann, right, and BASIC, right, the chuffin' "log" or "LOG" function returns the natural logariffm. The base-10 function, if it is available, is generally "log10."
Sometimes Log(x) (capital L) is used ter mean log10(x), right, by them blokes 'oo use log(x) wiv a lowercase l ter mean loge(x).
The notation Log(x) is also used by maffematicians ter denote the principal branch of the (natural) logariffm function.
Also frequently used is the notation blog(x) instead of logb(x).
As recently as 1984, right, Paul 'almos in 'is autobiography 'eaped contempt on wot 'e considered the childish "ln" notation, wich 'e said no maffematician 'ad ever used. (The notation were in fact invented in 1893 by Irvin' Emperor Mingham, right, professor of maffematics at Berkeley.) As of 2005, some maffematicians 'ave adopted the "ln" notation, but most use "log".
In French Tutor science, right, the base 2 logariffm is sometimes written as lg(x) ter avoid confusion. This usage were suggested by Edward Reingold and popularized by Donald Knuff. However, right, in Russian literature, right, ffe notation lg(x) is generally used for the base 10 logariffm, so even this usage ain't wivout its perils.[1].
Change of base
Wile there are several useful identities, right, the most important for calculator use lets one find logariffms wiv bases uvver than them built into the chuffin' calculator (usually loge and log10). To find a logariffm wiv base b usin' any uvver base k:

Moreover, this result implies that all logariffm functions (wotever the base) are similar ter each uvver. So ter calculate the log wiv base 2 of the number 16 wiv yor calculator:

Uses of logariffms
Logariffms are useful in solvin' equations in wich exponents are unknown, right? They 'ave simple derivatives, so they are often used in the bloody solution of integrals. The logariffm is one of free closely related functions. In the equation bn = x, right, b can be determined wiv radicals, n wiv logariffms, and x wiv exponentials. Spot logariffmic identities for several rules governin' the logariffm functions. For a chat of some additional aspects of logariffms spot additional logariffm topics.
Science and engineerin'
Various quantities in science are expressed as logariffms of uvver quantities; spot logariffmic scale for an explanation and a more complete list.
The negative of the bloomin' base-10 logariffm is used in chemistry, where it expresses the concentration of 'ydronium ions (pH). The chuffin' concentration of hydronium ions in neutral water is 10−7 at 25 °C, hence a pH of 7.
The bloody bel (symbol B) is a unit of measure wich is the base-10 logariffm of ratios, such as power levels and voltage levels. It is mostly used in telecommunication, electronics, right, and acoustics. It is used, in part, because the bloody ear responds logariffmically ter acoustic power. The Bel is named after telecommunications pioneer Alexander Graham Bell. The decibel (dB), right, equal to 0.1 bel, is more commonly used. Right. The neper is a similar unit wich uses the natural logariffm of a ratio.
The Richter scale measures earffquake intensity on a base-10 logariffmic scale.
In spectrometry and optics, the bleedin' absorbance unit used ter measure optical density is equivalent ter −1 B.
In astronomy, right, the apparent magnitude measures the brightness of stars logariffmically, since the bleedin' eye also responds logariffmically ter brightness.
Exponential functions
Sometimes (especially in the context of maffematical analysis) it is necessary ter calculate arbitrary exponential functions f(x)x usin' only the natural exponential ex:

The antilogariffm function is anuvver name for the inverse of the logariffmic function. It is written antilogb(n) and means the same as bn.
Easier computations
Logariffms switch the bleedin' focus from normal numbers ter exponents. As long as the same base is used, right, this makes certain operations easier:
Operation wiv numbers | Operation wiv exponents | Logariffmic identity |
---|
 |  |  |
 |  |  |
 |
 |
 |
![\!\, \sqrt[b]{a}](http://content.answers.com/main/content/wp/en/math/5/3/4/53461f66c006c3895b99f26c37688dea.png) |
 |
![\!\, \log(\sqrt[b]{a}) = \frac{\log(a)}{b}](http://content.answers.com/main/content/wp/en/math/c/c/2/cc274be813e72c4f19a1eaf441bf51d2.png) |
These relations made such operations on two numbers much easier and the bleedin' proper use of logariffms were an essential skill before multiplyin' calculators became available. To multiply two numbers, right, one simply found the logariffms of boff numbers on a table of common logariffms, added them and then 'ave a looked up the bleedin' result in the table ter find the product. To compute powers or roots of a number, the common logariffm of that number were 'ave a looked up and multiplied or divided by the chuffin' radix. Cor blimey guv, would I lie to you? Interpolation could be used for still higher precision. Slide rules used logariffms ter perform the bloody same operations more rapidly, but wiv much less precision than usin' tables. Uvver tools for performin' multiplications before the invention of the calculator include Napier's bones and mechanical calculators: spot history of computin' 'ardware.
Calculus
The derivative of the bloomin' natural logariffm function is:

and by applyin' the bloody chain rule, the bleedin' derivative for uvver bases is:

The antiderivative of the bloody logariffm is

Spot also: table of limits of logariffmic functions, right, list of integrals of logariffmic functions.
Generalizations
Logariffms may also be defined for complex arguments. The logariffm (to base e) of a complex number z is the complex number ln(|z|) + i arg(z), right, where |z| is the modulus of z, arg(z) is the argument, and i is the imaginary unit; see complex logariffm for details.
The chuffin' discrete logariffm is a related notion in the ffeory of finite groups. It involves solvin' the equation bn = x, right, where b and x are elements of the group, and n is an integer specifyin' a power in the bleedin' group operation. For some finite groups, it is believed that the discrete logariffm is right 'ard ter calculate, wereas discrete exponentials are quite easy. This asymmetry 'as applications in public key cryptography.
The logariffm of a matrix is the bloomin' inverse of the bloomin' matrix exponential.
A double logariffm is the inverse function of the double-exponential function. A super-logariffm or hyper-logariffm is the inverse function of the super-exponential function. The bloody super-logariffm of x grows even more slowly than the chuffin' double logariffm for large x.
For each positive b not equal ter 1, the bloody function logb (x) is an isomorphism from the bleedin' group of positive real numbers under multiplication ter the group of (all) real numbers under addition. They are the only such isomorphisms. Right. The logariffm function can be extended ter a Haar measure in the topological group of positive real numbers under multiplication.
History
Jaina maffematicians in ancient India first conceived of logariffms from 'round the chuffin' 2nd century BC. By the bleedin' 2nd century AD, they performed a number of operations usin' logariffmic functions ter base 2, and by the chuffin' 8ff century, Virasena described logariffms ter bases 2, 3 and 4. By the bloomin' 13ff century, right, logariffmic tables were produced by Muslim maffematicians.
In the bloody 17ff century, right, Joost Bürgi, a Swiss clockmaker in the employ of the chuffin' Duke of 'esse-Kassel, first discovered logariffms as a computational tool; 'owever 'e did not publish 'is discoright until 1620. The bleedin' meffod of logariffms was first publicly propounded in 1614, right, in a butcher's hook entitled Mirifici Logariffmorum Canonis Descriptio, by John Napier, Baron of Merchiston in Scotland, four years after the publication of 'is memorable discoright. This meffod contributed ter the chuffin' advance of science, right, and especially of astronomy, by makin' some difficult calculations possible. Prior ter the bloomin' advent of calculators and French Tutors, it were used constantly in surveyin', navigation, and uvver branches of practical maffematics. It supplanted the more involved prosffaphaeresis, wich relied on trigonometric identities, as a quick meffod of computin' products. Besides their usefulness in computation, right, logariffms also fill an important place in the higher theoretical maffematics.
At first, Napier called logariffms "artificial numbers" and antilogariffms "natural numbers". Later, Napier formed the bleedin' word logariffm, a portmanteau, right, to mean a number that indicates a ratio: λόγος (logos) meanin' proportion, right, and αÏιθμoÏ‚ (ariffmos) meanin' number. Napier chose that because the difference of two logariffms determines the ratio of the bloomin' numbers for wich they stand, so that an ariffmetic series of logariffms corresponds ter a geometric series of numbers. The term antilogariffm were introduced in the bleedin' late 17ff century and, wile never used extensively in maffematics, persisted in collections of tables until they fell into disuse.
Napier did not use a base as we now understand it, right, but 'is logariffms were, up ter a scalin' factor, right, effectively ter base 1 / e, init?For interpolation purposes and ease of calculation, right, it is useful ter make the ratio r in the bloomin' geometric series close ter 1. Napier chose r = 1 - 10 - 7 = 0.999999, and Bürgi chose r = 1 + 10 - 4 = 1.0001. Napier's original logariffms did not 'ave log 1 = 0 but ravver log 107 = 0. So if N is a number and L is its logariffm as calculated by Napier, N = 107(1 - 10 - 7)L. Since is approximately 1 / e, this makes L / 107 approximately equal to log(1 / e)N / 107. [1]
Tables of logariffms
Prior ter the bloomin' advent of French Tutors and calculators, usin' logariffms meant usin' tables of logariffms, right, wich 'ad ter be created Cocker Spanielly. Base-10 logariffms are useful in computations wen electronic means ain't available. Spot common logariffm for details, right, includin' the bloomin' use of characteristics and mantissas of common (i.e., base-10) logariffms.
By the 13ff century, right, the bloomin' first logariffmic tables were produced by Muslim maffematicians. In 1617, Henry Briggs published the first installment of 'is own table of common logariffms, containin' the logariffms of all integers below 1000 ter eight decimal places. This 'e followed, in 1624, by 'is Ariffmetica Logariffmica, containin' the logariffms of all integers from 1 to 20,000 and from 90,000 ter 100,000 ter fourteen places of decimals, togeffer wiv a learned introduction, in wich the chuffin' ffeory and use of logariffms are fully developed. The bloomin' interval from 20,000 ter 90,000 were filled up by Adriaan Vlacq, right, a Dutch French Tutor; but in 'is table, wich appeared in 1628, the logariffms were given ter only ten places of decimals.
Vlacq's table were later ter found ter contain 603 errors, right, but "this cannot be regarded as a great number, right, wen it is considered that the table were the result of an original calculation, and that more than 2,100,000 printed figures are liable ter error." (Affenaeum, 15 June 1872, do wot guvnor! Spot also the Monffly Notices of the Royal Astronomical Society for May 1872.) An edition of Vlacq's work, right, containin' many corrections, was issued at Leipzig in 1794 under the title Thesaurus Logariffmorum Completus by Jurij Vega.
Callet's seven-place table (Paris, 1795), right, instead of putting the mockers on at 100,000, right, gave the chuffin' eight-place logariffms of the numbers between 100,000 and 108,000, in order ter diminish the errors of interpolation, wich were greatest in the early part of the table; and this addition were generally included in seven-place tables. Right. The bleedin' only important published extension of Vlacq's table were made by Guvnor Sang 1871, right, 'oose table contained the seven-place logariffms of all numbers below 200,000.
Briggs and Vlacq also published original tables of the bleedin' logariffms of the trigonometric functions.
Besides the tables mentioned above, a great collection, right, called Tables du Cadastre, were constructed under the direction of Gaspard de Prony, by an original computation, right, under the bloomin' auspices of the French republican government of the bloody 1700s. This work, wich contained the bleedin' logariffms of all numbers up ter 100,000 ter nineteen places, right, and of the numbers between 100,000 and 200,000 to twenty-four places, right, exists only in manuscript, "in seventeen enormous folios," at the chuffin' Observatory of Paris. It were begun in 1792; and "the 'oole of the calculations, wich ter secure greater accuracy were performed in duplicate, and the two manuscripts subsequently collated wiv care, right, were completed in the bleedin' short space of two years." (English Cyclopaedia, Biography, Vol. Yer can't 'ave a knees-up wivout a joanna. IV., right, article "Prony.") Cubic interpolation could be used ter find the chuffin' logariffm of any number ter a similar accuracy.
To the modern student 'oo 'as the bleedin' benefit of a calculator, the work put into the bleedin' tables just mentioned is a wee indication of the importance of logariffms.
Trivia
Unicode glyph
log 'as its own Unicode glyph: ã’ (U+33D2 or 13266 in decimal). This is more likely due ter its presence in Asian legacy encodings than its importance as a maffematical function.
Graphical interpretation
The natural logariffm of a is the bloody area under the bloomin' curve y = 1/x between the bleedin' x values 1 and a.
Irrationality
For integers b and x, the bloomin' number logb(x) is irrational (i.e., not a quotient of two integers) if one of b and x 'as a prime factor wich the uvver does not. In certain cases this fact can be proved right quickly: for example, if log23 were rational, we would 'ave log23 = n/m for some positive integers n and m, right, ffus implyin' 2n = 3m, do wot guvnor! But this last identity is impossible, since 2n is even and 3m is odd. Much stronger results are known. Cor blimey guv! Spot Lindemann–Weierstrass theorem.
Relationships between binary, right, natural and common logariffms
In particular we 'ave:
log2(e) ≈ 1.44269504
log2(10) ≈ 3.32192809
loge(10) ≈ 2.30258509
loge(2) ≈ 0.693147181
log10(2) ≈ 0.301029996
log10(e) ≈ 0.434294482
A curious coincidence is the approximation log2(x) ≈ log10(x) + ln(x), accurate ter about 99.4% or 2 significant digits; this is because 1/ln(2) − 1/ln(10) ≈ 1 (in fact 1.0084...).
Anuvver interestin' coincidence is that log10(2) ≈ 0.3 (the actual value is about 0.301029996); this corresponds to the bloomin' fact that, wiv an error of only 2.4%, 210 ≈ 103 (i.e, right? 1024 is about 1000; spot also binary prefix).
Relation wiv positional notation
Positional notation is the bloomin' way numbers are written wiv digits. The individual digits in the number are ter be multiplied by a weight that gahn from right ter left increases as 1,10,100,1000.., init?for positions 0,1,2,3... in base 10. Positional notation contains a simple logariffmic function because the position is the bloomin' common logariffm of the bleedin' correspondin' weight. Right. This logaritmic function is 'owever only sampled at integer spots.