The word logarithm comes from the Greek (number and ratio) and is translated, therefore, as a ratio of numbers. The choice of the inventor (1594) of logarithms by J. Neper of this name is explained by the fact that logarithms arose when two numbers were compared, one of which is a member of an arithmetic progression, and the other is geometric. Logarithms with base e Speidel (1619), who compiled the first tables for the function ln x.Name of later origin natural (natural) is explained by the "naturalness" of this logarithm. N. Mercator (1620-1687), who proposed this name, discovered ln x - this is the area under the hyperbola y \u003d 1 / x. He also suggested the name hyperbolic.
N. Mercator
.During the 16th century, the volume of work associated with conducting approximate calculations in the course of solving problems, and primarily astronomy problems, has sharply increased, which has direct practical application (in particular, when determining the position of vessels by the stars and by the Sun). The biggest problems arose when performing multiplication and division operations. Attempts to partially simplify these operations by reducing them to addition did not bring much success. Therefore, the discovery of logarithms, which reduces multiplication, division of numbers to addition, subtraction of their logarithms, lengthened, according to Laplace, the life of calculators.
Logarithms came into practice unusually quickly. The inventors of logarithms did not confine themselves to the development of a new theory, but a practical tool, the table of logarithms, was created, which dramatically increased the productivity of calculators. We add that just 9 years after the publication of the first tables, the English mathematician D. Gunter invented the first slide rule, which became a working tool for many generations. (Until very recently, when before our eyes, electronic computing technology and the role logarithms, as a means of calculation is sharply reduced.) The first tables of logarithms were compiled independently of each other by the Scottish mathematician Napier (1550-1617) and the Swiss I. Burghi.
I. Burgi
Napier's tables, published under the titles "Description of the amazing table of logarithms" (1614) and "The structure of the amazing table of logarithms" (1619) included the values \u200b\u200bof the logarithms of sines, cosines and tangents for angles from 0 to 90 degrees in steps of one minute. Burghi prepared his tables of logarithms of numbers, apparently by 1610, but they were published in 1620, after the publication of Napier's tables, and therefore remained unnoticed.
One of the important ideas behind the invention of logarithms was already known. Stiefel and a number of other mathematicians drew attention to the fact that multiplication and division of a geometric progression
Corresponds to the addition and subtraction of indicators forming an arithmetic progression ..., - 3, -2, -1,0,1,2,3, ....
But this idea alone is not enough. For example, a "network" of integer powers of 2 is too rare; many numbers "remain without logarithms", so there was another idea: to raise numbers very close to one to a power. Noting that the powers
are close for large values \u200b\u200bof n, Napier and Burghi made a similar decision: Napier took the number
A Burgi-number
The further course of their reasoning and description of the calculation schemes is rather difficult to retell, both because there are many difficult details, and because the texts of the 16th century are rather vague. We only note that in fact Napier goes to the bottom
And Burgi - to the base
This did not change the essence of the matter, but made it possible to somewhat simplify the calculations and the tables themselves.
Thus, in essence, both inventors of the logarithm came to the conclusion that it is advisable to consider degrees of the form
where M is a very large number. Considering numbers of this kind leads to the number you know e which was defined as
It remains a little before the idea of \u200b\u200btaking the number e as the base of the logarithms (the base of the table of Burgi's logarithms coincides with the accuracy of the third decimal place with e, the base of the table of Napier's logarithms is close to 1 / e).
The first tables of decimal logarithms (1617) were compiled on the advice of Napier by the English mathematician H. Briggs.
Many of them were found using the approximate formula derived by Briggs, which is sufficiently accurate for large values \u200b\u200bof m and n in the form of powers of two: this made it possible for him to reduce the calculations to sequential extraction of square roots.
In compiling the tables of logarithms, an important role was played by the relation found by Napier and Burgi between the increments of x and y at an arbitrary point x for the function y \u003d logx. Taking away from the details of their presentation system, the main result can be expressed as follows:
, where k is some constant. If the base of logarithms is a power where n is a sufficiently large number, then
Tending to zero, we arrive at the differential equation y "\u003d 1 / x, the solution of which, as you know, is the function lnx + C. There is a system of presentation in which
from the very beginning it is defined as, i.e. - the area of \u200b\u200ba curved trapezoid bounded by a hyperbola, an abscissa and straight lines x \u003d 1 and
The only way to realize long-distance travel was navigation, which is always associated with the performance of large volumes of navigation calculations. Now it is difficult to imagine the process of grueling calculations when multiplying and dividing five-six-digit numbers "manually". the theologian by the nature of his main activity, doing trigonometric calculations at his leisure, guessed to replace the laborious procedure of multiplication with simple addition. He himself said that his goal was "to free yourself from the difficulty and boredom of calculations, which deter many from studying mathematics." The efforts were crowned with success - a mathematical apparatus called the system of logarithms was created.
So what is a logarithm? The basis of logarithmic calculations is a different representation of the number: instead of the usual positional system, as we are used to, the number A is represented as a power expression, where some arbitrary number N, called the base of the power, is raised to such a power n that the result is the number A. Thus , n is the logarithm of the number A to the base N. The choice of the base of the logarithms determines the name of the system. For simple calculations, the decimal system of logarithms is used, and in science and technology the system of natural logarithms is widely used, where the irrational number e \u003d 2.718 serves as the base. The expression that determines the logarithm of the number A in the language of mathematics is written as follows:
n \u003d log (N) A, where N is the base of the degree.
Decimal and natural logarithms have their own specific abbreviations - lgA and lnA, respectively.
In a calculation system that uses the calculation of logarithms, the main element is the conversion of a number to a power form using a table of logarithms in some base, for example 10. This manipulation does not present any difficulties. Further, the property of power numbers is used, which consists in the fact that when multiplied, their powers add up. In practice, this means that the multiplication of numbers with a logarithmic representation is replaced by the addition of their powers. Therefore, the question “what is a logarithm”, if we continue it to “why do we need it”, has a simple answer - to simplify the procedure for multiplying and dividing multidigit numbers - after all, addition “in a column” is much simpler than multiplication “in a column”. Who does not believe - let him try to add and multiply two eight-digit numbers.
The first tables of logarithms (based on base c were published in 1614 by John Napier, and a completely error-free version, including tables of logarithm decimal, appeared in 1857 and is known as Bremiker's tables. The use of logarithms with base in the form is due to the fact that the number e is quite simple get through the Taylor series, which is widely used in integral and
The essence of this computing system is contained in the answer to the question "what is a logarithm" and follows from the basic logarithmic identity: N (base of the logarithm) n equal to the logarithm of the number A (logA) is equal to this number A. At the same time, A\u003e 0, i.e. ... The logarithm is determined only for positive numbers, and the base of the logarithm is always greater than 0 and not equal to 1. Based on the foregoing, the properties of the natural logarithm can be formulated as follows:
- The domain of the natural logarithm is the entire number axis from 0 to infinity.
- ln x \u003d 0 - a consequence of the well-known relation - any number in the zero degree is equal to 1.
- ln (X * Y) \u003d ln X + lnY - the most important property for computational manipulations is the logarithm of the product of two ramen numbers to the sum of the logarithms of each of them.
- ln (X / Y) \u003d ln X - lnY - the logarithm of the quotient of two numbers is equal to the difference between the logarithms of these numbers.
- ln (X) n \u003d n * ln X.
- The natural logarithm is a differentiable, upward convex function, with ln ’X \u003d 1 / X
- log (N) A \u003d K * ln A - the logarithm in any positive base other than the number e differs from natural only by a coefficient.
Now every schoolchild knows what a logarithm is, but thanks to progress in the field of applied computing technology, the problems of computing work are a thing of the past. Nevertheless, logarithms, already as a mathematical tool, are used to solve equations with unknowns in the exponent, in expressions for finding time
(from the Greek λόγος - "word", "relation" and ἀριθμός - "number") numbers b by reason a (log α b) is called such a number cand b= a c, that is, log α b=c and b \u003d a c are equivalent. The logarithm makes sense if a\u003e 0, and ≠ 1, b\u003e 0.
In other words logarithm numbers b by reason ais formulated as an indicator of the degree to which the number must be raised ato get the number b(only positive numbers have a logarithm).
This formulation implies that the computation x \u003d log α b, is equivalent to solving the equation a x \u003d b.
For example:
log 2 8 \u003d 3 because 8 \u003d 2 3.
We emphasize that the above formulation of the logarithm makes it possible to immediately determine logarithm value, when the number under the sign of the logarithm is some degree of the base. And in truth, the formulation of the logarithm makes it possible to prove that if b \u003d a c, then the logarithm of the number b by reason a is equal from... It is also clear that the topic of logarithm is closely related to the topic degree of number.
Calculation of the logarithm is called by taking the logarithm... Taking the logarithm is a mathematical operation of taking the logarithm. When taking the logarithm, the product of the factors is transformed into the sum of the terms.
Potentiation is a mathematical operation inverse to logarithm. In potentiation, the given base is raised to the power of the expression over which the potentiation is performed. In this case, the sums of the members are transformed into the product of the factors.
Real logarithms with bases 2 (binary), e Euler's number e ≈ 2.718 (natural logarithm) and 10 (decimal) are used quite often.
At this stage, it is advisable to consider samples of logarithmslog 7 2 , ln √ 5, lg0.0001.
And the entries lg (-3), log -3 3.2, log -1 -4.3 do not make sense, since in the first of them a negative number is placed under the sign of the logarithm, in the second - a negative number at the base, and in the third - a negative number under the sign of the logarithm and one at the base.
Conditions for determining the logarithm.
It is worth considering separately the conditions a\u003e 0, a ≠ 1, b\u003e 0 under which definition of the logarithm. Let's consider why these restrictions are taken. An equality of the form x \u003d log α b , called the basic logarithmic identity, which directly follows from the definition of a logarithm given above.
Let's take the condition a ≠ 1... Since one is equal to one to any degree, the equality x \u003d log α b can exist only when b \u003d 1but log 1 1 will be any real number. To eliminate this ambiguity, we take a ≠ 1.
Let us prove the necessity of the condition a\u003e 0... When a \u003d 0 according to the formulation of the logarithm, it can only exist for b \u003d 0... And accordingly then log 0 0can be any nonzero real number, since zero in any nonzero degree is zero. To eliminate this ambiguity, the condition a ≠ 0... And when a<0 we would have to reject the analysis of rational and irrational values \u200b\u200bof the logarithm, since a degree with a rational and irrational exponent is defined only for non-negative grounds. It is for this reason that the condition is stipulated a\u003e 0.
And the last condition b\u003e 0 follows from the inequality a\u003e 0since x \u003d log α b, and the value of the degree with a positive base a always positive.
Features of logarithms.
Logarithms characterized by distinctive features, which led to their widespread use to significantly facilitate painstaking calculations. In the transition "to the world of logarithms" multiplication is transformed into a much easier addition, division into subtraction, and exponentiation and root extraction are transformed, respectively, into multiplication and division by an exponent.
The formulation of logarithms and a table of their values \u200b\u200b(for trigonometric functions) were first published in 1614 by the Scottish mathematician John Napier. Logarithmic tables, magnified and detailed by other scientists, were widely used in scientific and engineering calculations, and remained relevant until electronic calculators and computers were used.
In the sixteenth century, seafaring developed rapidly. Therefore, observations of celestial bodies were improved. To simplify astronomical calculations, logarithmic calculations arose in the late 16th and early 17th centuries.
The value of the logarithmic method lies in reducing multiplication and division of numbers to addition and subtraction. Less time consuming actions. Especially if you have to work with multi-digit numbers.
Burgi method
The first logarithmic tables were compiled by the Swiss mathematician Jost Burgi in 1590. The essence of his method was as follows.
To multiply, for example, 10,000 by 1,000, it is enough to count the number of zeros in the multiplier and the multiplier, add them (4 + 3) and write down the product of 10,000,000 (7 zeros). The factors are integer powers of 10. When multiplying, the exponents are added together. Division is also performed. It is replaced by subtracting exponents.
Thus, not all numbers can be divided and multiplied. But there will be more of them if we take a number close to 1. For example, 1.000001.
That is what the mathematician Jost Burgi did four hundred years ago. True, he published his work "Tables of arithmetic and geometric, together with a solid instruction ..." only in 1620.
Jost Burgi was born on February 28, 1552 in Liechtenstein. From 1579 to 1604 he served as court astronomer for the Landgrave of Hesse-Kassel Wilhelm IV. Later at the Emperor Rudolph II in Prague. A year before his death, in 1631, in Kassel. Burghi is also known as the inventor of the first pendulum clock.
Napier's tables
In 1614, John Napier's tables appeared. This scientist also took a number close to one as a base. But it was less than one.
Scottish Baron John Napier (1550-1617) studied at home. He loved to travel. Visited Germany, France and Spain. At the age of 21 he returned to the family estate near Edinburgh and lived there until his death. He was engaged in theology and mathematics. He studied the latter from the works of Euclid, Archimedes and Copernicus.
Decimal logarithms
Napier and the Englishman Brigg came up with the idea of \u200b\u200bcompiling a table of decimal logarithms. Together, they began the work of recalculating Napier's tables previously compiled. After Napier's death, Brigg continued it. He published the work in 1624. Therefore, decimal is also called brigade.
The compilation of logarithmic tables required many years of laborious work from scientists. On the other hand, the labor productivity of thousands of calculators who used the tables compiled by them increased many times over.
