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THE LOGARITHMS

INDEX

1. John Napier

2. Joost Bürgi

3. Henry Briggs

4. Leonhard Euler

5. Logarithm Map

6. Properties of logarithms

7. Two special logarithms

8. Laws of exponents and logarithms

9. Application map of logarithms

10. Applications and curiosities of logarithms

School year 2006-07

Class 3 C Teacher : Adriana Sileoni

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1. John Napier (1550-1617) John Napier was born in Merchiston Tower, Edinburgh, in 1550, the son of Archibald

Napier, the Master of the Mint in Scotland. Young John Napier entered St-Salvator's

College, University of St-Andrews, at the age of 13. It is rumored that during his youth,

Napier also studied in France, Germany, the Netherlands, and Italy. He only returned to

Scotland in 1571, to marry his first wife, Elizabeth, with whom he had a son. Elizabeth died

that same year, and in 1572, Napier married his second wife, Agnes, with whom he had

five sons and five daughters.

Most of the Napier family estates had been left into John's care, and he built himself a

castle, in which he and his family took up residence. Napier spent much of his time tending

to his estates, inventing new ways to fertilize the soil in order to grow more bountiful crops,

and maintaining greener grass. Napier, also referred to as the "Marvelous Merchiston",

may have come from a wealthy family, but he was a man of many talents, decent values

and high intelligence who truly deserved all he was awarded. 18th century philosopher

David Hume later wrote about Napier, saying he was a "person to whom the title of a great

man is more justly due than to any other whom his country ever produced".

Like many of history's great thinkers, Napier was very interested in religion, on both

scientific and philosophical terms. A fanatic of theology since his university days, John

Napier, a protestant like his father before him, had been born in a time when Scotland was

divided in two by a religious conflict between Roman Catholism (enforced by Mary Queen

of Scots) and Protestantism. In 1593, Napier published "A Plaine Discovery of the Whole

Revelation of St. John", a work in which he claimed his calculations based on the Book of

Revelations revealed to be the "Antichrist", going as far as predicting the coming of the

Apocalypse, some 100 years in the future.

Fortunately for us, Napier's "hobby", yielded much more accurate results. Little time was

left to the Theologian was spent studying mathematics, and developing new tools and

mnemonic formulas to make the process of calculation easier. Perhaps it was Napier's

lack of free time which led to his invention of Napier's roots, but there is no doubt his

greatest achievement was the publication of A Description of the Admirable Table of

Logarithms in 1614 (Mirifici logarithmorum canonis descriptio). While logarithms had

already been invented by Swiss mathematician Jost Burgi, it was Napier's work which

brought them into the public eye. Napier also claimed he had been contemplating

logarithms as early as 1694, and many of the era's most important scientists, including

astronomer Tycho Brahe were waiting in anticipation for his work to be published.

John Napier's work inspired many mathematicians, including Henry Briggs, professor of

geometry at Gresham College, and later on at Oxford. Following two visits to Edinburgh to

consult with Napier, Briggs devised a system of common logarithms with a decimal base.

In 1624, Briggs published Logarithmic Arithmetic (Arithmetica Logarithmica), which

contained the logarithms of 30000 natural numbers computed to 14 places. Napier himself

had been influenced by the work of Flemish scientist Simon Stevinus who, in 1585, had

developed a decimal fraction system; Napier eliminated the use of notation to indicate

fractional position, and he popularized the use of the decimal point.

In addition to the Mirifici logarithmorum canonis descriptio, Napier also published The Art

of Logic (De arte logistica), Rabdology, Study of Divining Rods (Rabdoligae seu

numerationis per vigulas libri duo), in which he explains the usage of Napier's roots, and

also The Art of Logic (De arte logistica), (1573 but not published until 1839).

Like many scientists, Napier took it upon himself to design machines to be used in warfare.

He produced a list of “devices for sailing under water with divers, other devises and

stratagems for harming the enemies”. These included a giant mirror which would act to

reflect the sun's rays upon incoming ships, burning them the same way a magnifying glass

is used to burn a hole in paper; he also dreamt up machines that would go underwater, to

protect against incoming vessels, and tank-like machines that could fire in all directions at

once.

Napier died on April 4th, 1617, in Edinburgh, Scotland, probably from complications arising

from his being afflicted with gout. The whereabouts of his remain are to day uncertain.

John Napier was indeed a very important man, important for Scotland, and the world. His

continuous quest for efficiency and his ingenious ways have no doubt eased the tasks of

mathematicians and scientists in for hundreds of years after his passing.

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2. Joost Bürgi (1552-1632) Born in Switzerland, Bürgi acquired considerable skills in mathematics and astronomical

observations and computations. In 1579, he became clockmaker and instrument-maker at

the Kassel observatory, the first to be fitted with a revolving dome. His early reputation as

instrument-maker prompted an invitation to Prague from the scientists of the imperial

court. He accepted in c. 1603, becoming clockmaker to Rudolf II (1552-1612). In Prague,

he helped Johannes Kepler (1571-1630) to perform calculations on the astronomical data

by Tycho Brahe (1546-1601). Bürgi built astronomical instruments distinguished for their

precision and beauty. He also crafted one of the first four-tip proportional compasses and

an unusual "theodolite" for perspective drawing, with a treatise on its use, now lost. Some

of his inventions were reworked and published in the writings of his brother-in-law

Benjamin Bramer (1588-1652). His most important mathematical contribution was a set of

log tables, which he composed before the end of the sixteenth century but did not publish

until 1620.

The Swiss watchmaker Joost Bürgi, maker of astronomical instruments and an

indefatigable computer and assistant to Johannes Kepler, Imperial astronomer in Prague,

also conceived a system of logarithms to facilitate the multiplication of large numbers.

Where Napier decided on powers of (1-10^7 ),Bürgi made the better choice,

(1+10^-4),

this made his power indices increase as the power numbers increased, while Napier`s

decreased.

There was another difference between the work of the two men: where Napier multiplied

his power by 10^7, Bürgi chose 10^8 he also multiplied his logarithms by 10 in his tables.

N= 10^8(1+10^-4)^L.

Bürgi called 10L the "red numbers" corresponding to the " black number" N. If all black

numbers are divided by 10^4, we obtain what is nearly a system of logarithms to the base

e. This also has something to do with natural logarithms. The official priority belongs to

Napier because of his publishing date, 1614, six years before Bürgi. It may be that he had

begun his work in 1588, or even 1584. Napier is reported to have discussed his results

with Brahe in 1594. So all we know is that they did this work independent from each other.

John Napier, Henry Briggs, and Joost Bürgi invented a good system that many people

before us, and sure many people after us will have a lot use for. I think many people are

very thankful for what he did to mathematics. Now you can find the correct answer to many

questions, while before Briggs fantastic system, they had to use a lot more time to find the

correct answer. Even if many student including us have hard time with logarithms, we

probably would have had bigger problems without them.

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3. Henry Briggs (1556-1631) Henry Briggs was a British mathematician, famous to have introduced the common

logarithms (in base 10) departing from the natural logarithms introduced by Nepero and to

effectively have contributed to their diffusion.

He was born at Warley Wood, next to Halifax, in West Yorkshire England. He was

admitted in 1577 St John's College in Cambridge and graduated in 1581. In 1588 he was

chosen Fellow of St. John's. In 1592 he became teacher of physics founded by Dr.

Thomas Linacre; he also held some lessons of mathematical matter. During this period he

also interested him of navigation and of astronomy, in collaboration with Edward Wright. In

1596 he became the first teacher of geometry in the just founded Gresham College in

London; here he held lessons for almost 23 years and he made of the Gresham College a

primary centre of the mathematics English; particularly Briggs defended the ideas of

Keplero.

In this period Briggs came in possession of a copy of the Mirifici Logarithmorum Canonis

Descriptio of Nepero; it struck his imagination and realized the potentialities of the

calculations through the use of the logarithms in astronomy and in the navigation; for this

he wrote and published in 1602 "A table to find the height of the pole, the declination

magnetic date" and in 1610 "Tables for the improvement of the navigation". In his lessons

at the Gresham College he proposed the change of the scale of the logarithms from the

base 1 / e that Nepero had introduced in his essay, to one in which the unity is the

logarithm of the relationship of 10 to 1. He wrote him and suggested him to change the

scale. Briggs introduced the logarithms in base 10 because he was looking for somehow

to find a remedy to the complex calculations that had to face with the natural logarithms of

Nepero. In fact the logarithms that Nepero invented are different from the logarithms that

we use today, in how much the fact that log1 doesn't equalize 0 is and this involves

notable complications making these less convenient to use in comparison to today's

logarithms. The person responsible for the change from natural logarithms to those with 10

was Briggs, even if the idea came from the discussions between Briggs and Nepero.

Briggs hed to write the tables because Nepero did not choose undertake a so hard job for

health and for other valid reasons but the idea that the logarithms had to have base 10 it

was of both and Briggs built the tables thanks also to the aid of Nepero.

Briggs was active in many circles and was often listened to after his opinions in the field of

the astronomy, of the surveying, of the navigation and other activities as the mining

extraction, approaching the theoretical mathematics to practical activity. In this period

Briggs made investments in the London Company, which made suppose that he was

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rather wealthy. In 1616 Briggs visited Nepero to Edinburgh to discuss the changes he

suggested to the logarithms of Nepero. The following year again visited him for analogous

reasons. During these meetings the change proposed by Briggs was welcomed and the

two mathematicians reached a system of logarithms in which the logarithm of 1 pits 0 and

the logarithm of 10 pits 1. Á. From his second visit to Edinburgh, in 1617, Briggs published

the first table of the logarithms of the numbers from 1 to 1000. He was named in 1619

Teacher of Geometry at Oxford and in July 1620 he gave the resignations from teacher to

the Gresham College. Immediately after his transfer in Oxford he was named Teacher of

the Arts. In 1622 he published a brief essay on "north-western Passage to the seas of the

South, through the continent of Virginia and the Bay of Hudson."

In 1624 he continued his Arithmetica Logarithmica. Besides he completed a table of the

logarithms of the sine and the shares for every cent of degree up to the 14a decimal figure,

with a table of the natural sinus up to the 15a decimal figure and the corresponding shares

and secant up to the decimal figure. All these jobs were printed to Gouda in 1631 and

published in 1633 with the title of British Trigonometry. This work was probably the

succession of his "Introduction to the Logarithms" (First Logarithmorum Chilias) of 1617,

that he introduced a brief explanation of the logarithms and a 1000 long table was

calculated up to the 14a decimal figure.

Briggs also discovered, without demonstration, the binomial theorem. Briggs was buried in

the Chapel of the Merton College in Oxford. Dr Smith, in his Grapevine of the Teachers of

Gresham, describes him as a man of great honesty, that despite the wealth he was

satisfied of his possessions, preferring the studies to a luxurious life.

In the first years of the 1600 John Napier and Henry Briggs worked to a method to express

the positive numbers as powers of 10:

3=100,47712 5=100,69897 15=101,17609

These exponents in base 10 were called logarithms. Briggs published a logarithmic table

that reproduced all the logarithms in base 10. The following small example of the table that

Briggs wrote:

We can find examples of logarithms in a lot of things that surround us and also in our

body. All of our senses are noticed by our brain in logarithmic intensity. This particularity of

our senses allows us to make a lot of things. The logarithmic answer of our sight to a

bright signal allows us to see the stars in a dark night without remaining dazzled by a

landscape illuminated by the Sun in full day. The logarithmic answer of the hearing allows

us to listen to the rustle of the leaves in a day of light breeze but also to hear without

damages the roar of an airplane that takes off.

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4. Leonhard Euler (1707-1783) Leonhard Euler, famous in Italy like Euler, is considered the most important mathematician

of Enlightenment. He was a student of Johann Bernoulli, he is famous to be the most

prolific of all the times and has supplied crucial contributions in different areas: infinitesimal

analysis, special functions, mechanics ration, mechanical celestial, theory of the numbers,

theory of the graphs.

Euler was the greatest supplier of “mathematical denominations”, offering his name to one

impressive amount of formulas, theorems, methods, criteria, relations, equations. In

geometry: the relative circle, straight and points of Euler to the triangles, plus the relation

of Euler, that he studied the circle circumscribed to a triangle; in the theory of the numbers:

the criterion of Euler, the pointer of Euler, the identity of Euler, the conjecture of Euler; in

the mechanics: the angles of Euler, the critical cargo of Euler (for instability); in the

analysis: the constant of Euler-Large mask; in logic: the diagram of Euler-Venn; in the

theory of the grafi: (of new) the relation of Euler; in algebra: the method of Euler (relative to

the solution of the equations of fourth degree); in the differential calculus: the method of

Euler (regarding the equations differentiates).

Other objects are connected to Euler “the Eulerian” adjective,: the Eulerian cycle, the

Eulerian graph, the Eulerian function or function beta, and that one of second species or

range function, the Eulerian numbers (different from the Numbers of Euler).

Even if he was mostly a mathematician gave important contributions to the classic

mechanics and celestial physics. As an example he developed the equation of bundle of

Euler-Bernoulli and the equations of Euler-Lagrange. Moreover he determined the orbits of

many comets.

Euler was born in Basel, son of Paul Euler, protestant priest, and Marguerite Brucker. He

had two sisters Anna Maria and Maria Magdalena. After the birth of Leonhard, the family

moved to Riehen, where Euler spent great part of infancy. Paul Euler was friend of the

family Bernoulli, and Johann Bernoulli, one of most famous mathematicians of Europe, he

had much influence on Leonhard. Euler entered in the University of Basel at the age of 13.

He graduated himself in philosophy. In that time he received also lessons of mathematics

from Johann Bernoulli, that he had discovered his enormous talent. The father of Euler

wanted that he became a theologian and he studied Greek and Hebrew. Fortunately

Bernoulli convinced the father of Euler to the mathematic career. In 1726 Euler completed

his doctorate on the propagation of the sound and in 1727; he participated to the Grand

Prix of the French Academy of sciences. The problem of that year regarded the better way

to arrange the trees on one ship. He arrived second after Pierre Bouguer he was

recognized as the father of the naval architecture.

In those years the two sons of Johann Bernoulli, Daniel and Nicolas worked in the Imperial

Academy of sciences of Saint Pietroburgo. In 1726, Nicolas died and Daniel taught

mathematics and physics, leaving vacant his chair in medicine. For this Euler accepted.

Euler arrived in Russia in 1727 after. Little time he passed in the medicine department

then he passed in mathematics. In those years he lived with Daniel Bernoulli with whom

he started an intense mathematical collaboration.

The stressful Russian life tired Euler who loved a calmer life. He was offered a place at

the Academy of Berlin from Federico. Euler accepted and left Berlin in 1741. He lived in

Berlin for 25 years.

Euler introduced many notations in use still today: between these, f (x) for the function, he

worked the trigonometric functions like sine and cosine, and the Greek letter Σ for the

summary one. First he used the letter e in order to indicate the base of the natural

logarithms, a real number that exactly are called also number of Euler, and the letter i in

order to indicate the imaginary unit. The use of the Greek letter π in order to indicate pi

Greek. .

First of all Euler introduced the concept of the function, the use of the exponential function

and the logarithms.

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5. Logarithm Map

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6. Properties of Logarithms In the previous section, we introduced the logarithmic functions, log b, as the inverses of

the exponential functions expb(x) = bx. We saw that, although they cannot be given

defined by any algebraic formula, they needed to solve exponential equations. In this

section, we will discuss the algebraic properties of logarithmic functions, which we will use

to calculate logarithms and solve exponential equations. We will see that these properties

follow the rules of exponents. We will begin, however, by discussing two very important

logarithmic functions, loge and log10.

7. Two Special Logarithms

You may be surprised to find that, despite the fact that there are many different logarithmic

functions, your calculator has buttons for only two of them, namely, loge and log10. In

fact, you may not even have recognized them, since we normally use a slightly different

notation for these two, special logarithms. Since we use a "base 10" number system (i.e.,

numbers are written in a way that implicitly uses powers of 10), it not surprising that we

should treat the base 10 logarithms in a special way. As we said before, base 10

logarithms were first calculated by Henry Briggs in 1616, to simplify astronomical

calculations. Probably because this logarithm was used so commonly, people began to

leave off the subscript. This can be seen as early as 1647 in the writings of William

Oughtred ( Florian Cajori, A History of Mathematical Notations, vol. 1, p. 193). That is,

If the base is not explicitly written, the base is assumed to be 10, so that log(x) = log10(x).

This is sometimes referred to as the "common" logarithm.

Notice that values for this "common" logarithm are fairly easy to estimate to at least one

correct digit. For example, since log10(2153) is the only solution to 10x = 2153, and 103 =

1000 < 2153 < 10000 = 104, we know that 3 < log10(2153) < 4. Since the value in the

middle of the first inequality was closer to the left than the right, we expect this to be true in

the second inequality as well. Therefore, we can estimate one digit log10(2153) » 3.

Similarly, since 10-1 = 0.1 > 0.03125 > 0.01 = 10-2, we know that -1 > log10(0.03125) > -

2, i.e., to one digit log10(0.03125) » -2. Note: If you are familiar with "scientific notation",

that is, where we write 2153 = 2.153 × 103 and 0.03125 = 3.125 × 10-2, you can see that

there is a close relationship between the value of the common logarithm of a number and

its "order of magnitude" (i.e., the size of the exponent in scientific notation).

While the common logarithm is easier to compute, we have seen that the exponential

function, ex, with base e = 2.718..., arises naturally in a wide variety of applications, from

banking to population growth. Thus:

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We call loge the "natural" logarithm and abbreviate it as "ln". That is, ln(x) = loge(x).

Although this logarithmic function was the first to be calculated, by John Napier in 1614, it

was quickly been over by the common logarithm commonly used. It wasn't until Calculus

become widely used that mathematicians eventually realized that this was, in fact, the

most "natural" logarithm to be used. The abbreviation, "ln", for the natural logarithm did

not develop until even later. One of the earliest known uses was only in 1893.

8. Laws of Exponents and Logarithms

Since logarithmic and exponential functions are inverses, it is not surprising that the

algebraic properties of exponential functions (expressed in the rules of exponents) should

lead to useful algebraic formulas for logarithms. Moreover, since all of the rules of

exponents are reflected in symmetry properties of exponential graphs, we can begin to see

the corresponding properties of logarithmic functions by looking at their graphs.

For example, we know that a horizontal shift in an exponential graph corresponds to a

vertical scaling. Since a logarithmic graph come from interchanging the horizontal and

vertical axes in an exponential graph, this would suggest that an horizontal scaling in a

logarithmic graph should correspond to a vertical shift. We can see this by comparing the

graphs of, log2(x) and log2(8x).

Looking at various points along the graph, such as:

x y =

log2(x)

y =

log2(8x)

1

2

4

8

0

1

2

3

3

4

5

6

you can see that log2(8x) = log2(x) + 3. You may notice that the number "3" is somewhat

special, since 8 = 23. That is, x = 3 is the solution to 8 = 2x, or, using logarithm notation,

log2(8) = 3. In particular, we have the equation: log2(8x) = log2(x) + log2(8) = log2(8) +

log2(x). This suggests that logarithms satisfy the following rule:

Logarithms take products to sums. That is, for any base, b > 0, and numbers x, y > 0,

logb(xy) = logb(x) + logb(y).

We can prove this very easily. From rules of exponents, we know that bx + y = bxby, that

is, exponential functions take a sum in the inputs to a product of the outputs. Reversing

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the roles of inputs and outputs, this implies that logarithmic functions take product in the

inputs to a sum of the outputs, which is exactly the rule of logarithms we have just stated.

If you would prefer a more algebraic proof, assign variable names to the values logb(x) and

logb(y), say, u = logb(x) and v = logb(y). By the definition of logarithms, we then know that

bu = x and bv = y. Multiplying the two equations, we get xy = bubv = bu + v. Applying the

definition of logarithms again he have gives logb(xy) = u + v = logb(x) + logb(y), as

expected.

As you might expect, this has a companion rule:

Logarithms take quotients to differences. That is, for any base, b > 0, and numbers

x, y > 0, log b(x/y) = log b(x) – log b(y).

The proof is almost identical to that of the previous rule, so we will omit it. However, this

leads directly to another important rule. Since exponential functions always take the value

1 for an input of 0, logarithmic functions always take the value 0 for an input of 1. This

means, if we take x = 1 in the quotient rule,

that logb(1/y) = log b(1) – log b(y) = 0 – log b(y) = -log b(y).

That is: Logarithms take reciprocation to negation.

That is, for any base, b > 0, log b(1/x) = -log b(x).

We can easily derive a related rule. If we let u = log1/b(x), then (1/b)u = x. Taking

reciprocals of both sides we have 1/x = b u, so that log b(1/x) = u = log1/b(x). That is.

For any base, b > 0, log 1/b(x) = log b(1/x).

Taken together, these two formulas imply that:

For any base, b > 0, log 1/b(x) = log b(1/x) = -log b(x).

In particular, a vertical flip of a logarithmic graph corresponds to a reciprocal of its base.

This corresponds to the fact, which we have already observed, that a horizontal flip in an

exponential graph corresponds to taking a reciprocal of the base. We can see this by

comparing the graphs of, log2(x) and log 1/2(x).

Notice how the following points are on the graphs:

x y =

log2(x)

y =

log1/2(x)

1

2

4

8

0

1

2

3

0

-1

-2

-3

so that log1/2(x) = -log2(x).

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Our next example illustrates one of the most important properties of logarithms, as

expressed by the following formula:

You can pull powers out of logarithms. That is, for any base, b > 0, positive number, x > 0,

and any number, y, log b(xy) = y log b(x).

Notice that this includes the previous formula on reciprocals as a special case: with y = -1,

this says that log b(x-1) = (-1)log b(x); by rules of exponents, this is the same as,

log b(1/x) = -log b(x). In the next section, we will see how this property will allow us directly

relate exponential and linear models. We will also show how this formula leads to the

"change-of-base" formula, which will allow us to compute all logarithmic functions in terms

of the natural and common logarithms (i.e., ln and log) on our calculators, and thus solve

exponential equations.

We can verify this fact graphically, by comparing the graphs of, log2(x) and log2(x3).

Notice how the following points are on the graphs:

x y =

log2(x)

y =

log2(x3)

1

2

4

0

1

2

0

3

6

so that log2(x3) = 3log2(x), as expected. We can also prove this specific case, by using the

product rule discussed earlier: log2(x3) = log2(xxx) = log2(x) + log2(x) + log2(x) = 3log2(x).

The general, algebraic proof follows as before. Let u = logb(x) so that bu = x. Taking

exponents of both sides we have xy = (bu)y = buy = byu. This implies that logb(xy) = yu = y

logb(x), as explained. Notice how we use the rule of repeated exponents to prove this.

This means that, although it may not seem so at first, this rule of logarithms is the

"reverse" of rule of repeated exponents.

We will now use this power formula to derive the most important formula of this section.

Remember how in the previous section, we suggested that logk(x) could be written as

ln(x)/ln(k). We can show that this is only one case of the "change-of-base" formula:

Any two logarithmic functions are multiples of one another. That is, for any bases, b and c

> 0, there is a constant, K, so that logb(x) = K logc(x). Specifically, logb(x) = logb(c) logc(x);

that is, K = logb(c). It is often more convenient to divide and write this as

Since ln = loge, we can now see that logk(x) = ln(x)/ln(k) is an example of the latter formula

with b = e and c = k.

Notice how the following points are on the graphs:

x y =

log2(x)

y =

log4(x)

1

2

4

0

1

2

0

0.5

1

so that log2(x) = 2 log4(x) = log2(4) log4(x), since 2 = log2(4) (i.e., 22 = 4).

The algebraic proof of this follows easily from the previous rule of logarithms. As usual,

we begin by letting u = logc(x) so that cu = x. Now apply logb to both sides to obtain logb(x)

= logb(cu) = u logb(c) = logc(x)logb(c) = logb(c)logc(x). Notice how we used the previous rule

of logarithms to obtain the second equality.

While this may seem like a pretty esoteric property of logarithms, it is actually one of the

most important. For instance, it justifies our statements about the relationship between

logarithmic and exponential graphs of different bases. More important, it says that the two

buttons on our calculator are enough. In fact, we need only one of them! For example, we

can calculate log5(20) as log10(20)/log10(5) » (1.30103)/(0.69897) » 1.86135.

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9. APPLICATION MAP OF LOGARITHMS

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10. Applications and curiosities of logarithms Wars are classified on the number of dead man and we talk about magnitude 3, 4 or 7 to

indicate the exponent that on 10 base, indicates the number of dead men approximately,

that is, in short, the logarithm according to 10 base of the number of the dead men.

Therefore if you listen to that a war has a double magnitude respect to a previous war, we

must be alarmed because as far as the numbers of dead men, that war had much more

than the double quantity. An example, a war of magnitude M = 3 will have little more than

a thousand victims, while a war of magnitude M = 6 will have more than a million of dead

men.

In finance field, the last work to find fiscal evaders, is connected with the logarithms and

the mathematician Mark Nigrini worked on it and used, it using the law in 1881 discovered

by the mathematician Simon Newcomb, then formalized by the physicist Frank Benford in

1938. Newcomb had noticed that in the library of the University tables of the logarithms, a

lot used, were more thorn at the beginning than the end and he deduced that the numbers

used in the calculations from his colleagues more often began with 1 than 2, with 2 than 3

ecc. From this observation he obtained an empiric law of distribution of the numbers used

from scientists that we can reassume in the formula:

Probability (that the first figure of the number is d)=

Where d indicates one of the figures from 1 to 9. Benford tested the formula using the

several number of numeric data. He noticed that not all the tables of data obey to the law:

the tidiest data, as an example to the table containing the squares of the numbers, do not

respect the law, while the most dishomogeneous tables respect it nearly completely.

Nigrini uses the law of Benford elaborating programs allowing suspicious numerical

distributions be found in the incomes tax return: in fact when a person tries to invent a

sequence of accidental numbers in order to simulate its

situation his financial situation, he obtains instead some

numbers correlated between them. In this way fiscal evaders,

produce incomes tax return that analyzed evidence

remarkable shunting lines from the law of Benford. Therefore

to know if the evader has honestly compiled the income tax

return it’s enough to control the frequency of the several used

figures in order to write the numbers.

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Moreover the logarithms, in astronomy, are used in the definition of Magnitude of a star.

The stellar Magnitude are logarithms and this in agreement with how much happens in our

eye where the answer to the bright impulses is logarithmic (and not linear). In this way we

can see both a dim light and a big lightning. The first one to speak about stellar Magnitude

was Ipparco of Nicea (300 a.C.). He defined the brightest stars those biggest, those just

seen in the sixth wideth. The stars of the second wideth were approximately 2 times and

half weaker those than before. Today a star of 1 magnitude is 100 times more brighter

than one of 6 magnitude so, if we want to know the exact relation of brightness between a

magnitude and the following one, we must divide 100 in 5 parts in geometric proportion, so

that the relation is constant between a part and with that one immediately previous. That

is to calculate the

Therefore taking this number as the base for some logarithms, that we will call stellar

logarithms, we write progression 1, 2,512; 6,310; 15,849; ...... The indicate numbers

represent the successive powers of 2,521

From this last sequence we can see that the following natural numbers that are used for

the magnitude are not other else that the logarithms (the exponents) that must elevate

base 2,512 to obtain the value of the brightness of one star. The logarithms are connected

with earthquakes too. The scale Richter, in fact, measure the magnitude of an earthquake

based on the amount of energy freed to the epicentre. It’s important to know that the used

scale is logarithmic because an earthquake of magnitude 8 is not doubly more disastrous

than one of magnitude 4. We work on exponents so, 108 represents 10000 x 104 that is

10000 times more disastrous!

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