Logarithms. History of origin.
What is a logarithm?
Logarithm a positive number b to base a, where a > 0,a ≠ 1, is called the exponent to which the number a must be raised to get b/
Logarithms are rhymes
Like words in music.
They make calculations easier -
No more difficult than twice two.
The word LOGARITHM comes from the Greek words - number and - ratio. translated as a ratio of numbers, one of which is a member of an arithmetic progression, and the other of a geometric progression. The word LOGARITHM comes from the Greek words - number and - ratio. translated as a ratio of numbers, one of which is a member of an arithmetic progression, and the other of a geometric progression.
LOGARITHM is a number that can be used to simplify many complex arithmetic operations. Using logarithms instead of numbers in calculations allows you to replace multiplication with the simpler operation of addition, division with subtraction, exponentiation with multiplication, and extraction of roots with division.
The concept of logarithms was first introduced by the English mathematician John Napier. Descendant of an old warlike Scottish family. He studied logic, theology, law, physics, mathematics, ethics. He was interested in alchemy and astrology. Invented several useful agricultural implements. In the 1590s, he came up with the idea of logarithmic calculations and compiled the first tables of logarithms, but published his famous work “Description of the Amazing Tables of Logarithms” only in 1614.
John Napier 1550-1617
The first tables of decimal logarithms were compiled in 1617 by the English mathematician Briggs. Many of them were derived using Briggs' formula.
The inventors of logarithms did not limit themselves to creating logarithmic tables; already 9 years after their development, in 1623, the English mathematician Gunter created the first slide rule. It has become a working tool for many generations. Nowadays we can find the values of logarithms using a computer. Thus, in the BASIC programming language, using the built-in function, you can find the natural logarithms of numbers.
Logarithmic ruler
"Logarithms are different..."
Briggs logarithm- the same as the decimal logarithm. Named after G. Briggs.
Decimal logarithm- logarithm to base 10. The decimal logarithm of a number is denoted lga.
Naper's logarithm- (named after J. Napier), the same as the natural logarithm.
Natural logarithm- logarithm, the base of which is Neper’s number e = 2.718 28... The natural logarithm of a number a is denoted by ln a.
John Napier ( 1550-1617)
Logarithms had the greatest influence on the development of astronomy. The successes of navigation in the Middle Ages led to a great demand for astronomical tables, the compilation of which required very complex calculations. The use of logarithmic tables greatly simplified and accelerated these calculations. According to the figurative expression of the French mathematician Laplace (1749-1827), the invention of logarithms, by reducing the work of the astronomer, extended his life.
The general definition of the logarithmic function and its broad generalization was given by Leonhard Euler.
In mathematics, logarithmic spiral
first mentioned in 1638
Rene Descartes.
Birds of prey circle their prey in a logarithmic spiral. The fact is that they see better if they look not directly at the prey, but slightly to the side.
Logarithmic spiral in nature
One of the most common spiders, when weaving a web, twists the threads around the center in a logarithmic spiral.
Application of logarithms
Music
The so-called steps of the tempered chromatic scale (12-sound) frequencies of sound vibrations are logarithms. Only the base of these logarithms is 2 (and not 10, as is customary in other cases). The piano key numbers are logarithms of the vibration numbers of the corresponding sounds.
Stars, noise and logarithms
The loudness of noise and the brightness of stars are assessed in the same way - on a logarithmic scale.
Psychology
By studying logarithms, scientists came to the conclusion that the magnitude of the sensation is proportional to the logarithm of the magnitude of irritation.
Why do we study logarithms?
Firstly, logarithms still allow us to simplify calculations today.
Secondly, from time immemorial, the goal of mathematical science was to help people learn more about the world around them, to understand its patterns and secrets.
Conclusion: logarithms are important components not only of mathematics, but also of the entire world around us, so interest in them has not waned over the years and they need to continue to be studied.
An important step in the study of logarithms was made by the Belgian mathematician Gregory of Saint-Vincent (1647), who discovered the connection between logarithms and areas limited by the arc of a hyperbola, the x-axis and the corresponding ordinates. The representation of the logarithm by an infinite power series was given by N. Mercator (1668), who found that In(1+x) = x Soon after, J. Gregory (1668) discovered the expansion ln This series converges very quickly if M = N + 1 and N is sufficient big; therefore it can be used to calculate logarithms. The works of L. Euler were of great importance in the development of the theory of the logarithm. He established the concept of logarithm as the inverse action of raising to a power.
LEONARD EULER ()
Thus, already in the middle of the 16th century. The fundamentals of the study of logarithms were developed. There was, however, a lack of useful, concrete methods for the widespread practical application of these fundamentals in computational mathematics; there was a lack of logarithmic tables based on a conscious idea. At the end of the 16th century. Simon Stevin published a table for calculating compound interest, the need for calculation of which was caused by the growth of commercial and financial transactions. As you know, the formula for compound interest is: A =a(1+(p/100))t where a is the initial capital, A is the accumulated capital after t years at P%. Stevin’s table contained the values of the expressions (1+(p/100))t, while (p/100) =r Stevin already expressed it in decimal fractions: 0.04; 0.05;..., which he first discovered in Europe. Stevin himself, oddly enough, did not notice that his tables could be used to simplify the corresponding calculations. However, one of his contemporaries, Burgi, saw this
The invention of logarithms at the beginning of the 17th century. closely related to the development in the 16th century. production and trade, astronomy and navigation, which required improvement of methods of computational mathematics. Increasingly, it was necessary to quickly perform cumbersome operations on multi-digit numbers; the results of actions had to be more and more accurate. It was then that the idea of logarithms was embodied, the value of which lies in reducing complex actions of the third stage (exponentiation and root extraction) to simpler actions of the second stage (multiplication and division), and the latter - to the simplest ones, to the actions of the first stage (addition and subtraction).
The first tables of logarithms were compiled independently of each other by the Scottish mathematician J. Napier () and the Swiss I. Burgi (1552 - 1632 (spent about 8 years on this work). The Englishman Henry Briggs () - developed a large table of decimal logarithms. English mathematics teacher John By 1620, Speidel compiled tables of natural numbers from 1 to London professor Edmund Tunter invented the logarithmic scale, the prototype of the slide rule Invention of logarithms.
Already in 1623, i.e. 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 became widespread and the role of logarithms as a means of calculation sharply decreased.
The term “LOGARITHM” was proposed by J. Napier; it arose from a combination of the Greek words logos (here relation) and arithmos (number), which meant “number of relations.” The term “natural logarithm” belongs to N. Mercator. The modern definition of logarithm was first given by the English mathematician W. Gardiner (1742). The logarithm sign, the result of the abbreviation of the word “LOGARITHM”, is found in various forms almost simultaneously with the appearance of the first tables [for example, Log in I. Kepler (1624) and G. Briggs (1631), log and in B. Cavalieri (1632, 1643)] . Historical reference
The first logarithmic tables were published in Russian in 1703. But in all logarithmic tables there were calculation errors. The first error-free tables were published in 1857 in Berlin, processed by the German mathematician K. Bremiker ()) 1. Kolmogorov A.N.. Algebra and the beginnings of analysis. Textbook for classes of general education institutions. M., “Enlightenment”, Algebra and the beginnings of analysis. Textbook for classes. Edited by Sh.A. Alimov et al. 11th ed. M.: Education, List of used literature
“Logarithm of a number” - Definition of logarithm. Finding the exponent from given values of the exponent and its base. Basic properties of the logarithm. A decimal is a logarithm whose base is 10. Properties of the logarithm. Basic logarithmic identity. Logarithms. Basic logarithmic identity. The concept of the logarithm of a number.
“Natural logarithm” - A logarithm to base e is called a natural logarithm. "Logarithmic Darts" Natural logarithms. Decimal logarithms are quite convenient for our needs. Calculate the area of the figure bounded by the lines y=0, x=1, x=e and the hyperbola. Write an equation for the tangent to the graph of the function y=lnx at the point x=e.
“Logarithmic functions” - Logarithmic function. Logarithm of the root. Logarithm of the degree. Properties of natural logarithms. Solutions of logarithmic equations. Function properties. The concept of logarithm. Logarithm of the product. Properties of logarithms. Transition from one indicator to another. Solving logarithmic inequalities. Solving logarithmic equations.
“Properties of logarithms” - Definition of a logarithm. If a>0 and a?1, x>0, y>0, p? R, then: Johann Heinrich Pestalozzi. 4. At what values of x does log5x exist; log3(x-7) ? 3. Formulate the basic properties of logarithms and calculate: log618 + log62 ; log553 ; log318 – log32 ; log2 log4 + log25 ; Counting and calculations are the basis of order in the head.
“Inventor of the logarithm” - Correct completion of some tasks. Raising to a power has two inverse effects. Why were logarithms invented? Ordnance. The definition of a logarithm can be written as follows: a log a b = b. Logarithms and their properties. Basic logarithmic identity. Correct solutions to examples. Logarithms were invented to speed up and simplify calculations.
“Logarithm Lesson” - Puzzle. Have you achieved your goal? Define logarithm. Logarithmic comedy. What else needs work? Computer independent work. Oral test - survey. Electronic test. Individual work. Lesson generalization on the topic “Logarithms”. Calculate: General solution. Solution.
Slide 1
Slide description:
Slide 2
Slide description:
Slide 3
Slide description:
Slide 4
Slide description:
Slide 5
Slide description:
Slide 6
Slide description:
The invention of logarithms Logarithms came into practice unusually quickly. The inventors of logarithms did not limit themselves to developing a new theory. A practical tool was created - tables of logarithms - which sharply increased the productivity of calculators. The first tables of logarithms were compiled independently of each other by the Scottish mathematician J. Napier (1550 - 1617) and the Swiss I. Burgi (1552 - 1632). Napier's tables, published in books entitled "Description of the amazing table of logarithms" (1614) and "The device of the amazing table of logarithms" (1619), included the values of logarithms of sines, cosines and tangents for angles from 0 to 90 in increments of 1 minute. Burgi 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 went unnoticed.
Slide 7
Slide description:
Slide 8
Slide description:
Slide 9
Slide description:
Portrait gallery Scottish mathematician, inventor of logarithms. Studied at the University of Edinburgh. Napier mastered the basic ideas of the doctrine of logarithms no later than 1594, but his “Description of the amazing table of logarithms”, in which this doctrine was set out, was published in 1614. This work contained a definition of the logarithm, an explanation of their properties, tables of logarithms of sines and cosines , tangents and applications of logarithms in spherical trigonometry. In “The Construction of a Surprising Table of Logarithms” (published in 1619), Napier outlined the principle of calculating tables.
Slide 10
Slide description:
Portrait gallery Archimedes' main works concerned various practical applications of mathematics (geometry), physics, hydrostatics and mechanics. In his work “Parabolas of Quadrature,” Archimedes substantiated the method of calculating the area of a parabolic segment, and he did this two thousand years before the discovery of integral calculus. In his work “On the Measurement of a Circle,” Archimedes first calculated the number “pi” - the ratio of the circumference to the diameter - and proved that it is the same for any circle.
Slide 11
Slide description:
Slide 12
To view the presentation with pictures, design and slides, download its file and open it in PowerPoint on your computer.
Text content of presentation slides: History of logarithms The term “logarithm” arose from a combination of the Greek words logos - ratio, ratio and arithmos - number and is literally translated as the ratio of numbers. Logarithms were discovered by Scottish mathematician John Napier in the early 17th century. Napier John (1550 – 1617), Scottish mathematician, inventor of logarithms. Napier is also the compiler of the first table of logarithms, which facilitated the work of calculators for many generations. The discovery of logarithms had a great influence on the development of applications of mathematics. The applications of exponential and logarithmic functions in a wide variety of fields of science and technology are endless, but logarithms were invented to facilitate calculations. More than three centuries have passed since the first logarithmic tables compiled by John Napier were published in 1614. They helped astronomers and engineers, reducing the time for calculations, and thus, as the famous French astronomer, mathematician and physicist Laplace said, “The invention of logarithms, by reducing the work of the astronomer, extended his life.” Slide rule (counting ruler), a counting tool for simplifying calculations, with the help of which operations on numbers are replaced by operations on the logarithms of these numbers. Designed for engineering and other calculations. Until recently, it was difficult to imagine an engineer without a slide rule in his pocket; which was invented ten years after the appearance of logarithms. It was invented by the English mathematician Gunther. It made it possible to quickly obtain an answer with an accuracy of three significant figures sufficient for an engineer. Now it has been squeezed out of engineering use by microcalculators. But without the slide rule, neither the first computers nor microcalculators would have been built. …Even the fine arts are nourished by it. Isn’t the musical scale a Set of advanced logarithms? From “Ode to the Exponential” The manifold uses of the exponential function inspired the English poet Elmer Bril and he wrote “Ode to the Exponential” We know that the exponential and logarithmic functions are mutually inverse. An exponential function is also called an exponent. Logarithms in art There were poets who did not devote odes to exponents and logarithms, but mentioned them in their poems. For example, in his poem, the poet Boris Slutsky wrote the lines Because the word is foam, Our rhymes fall. And greatness gradually Retreats into logarithms Boris Slutsky Musicians are rarely interested in mathematics; most of them have a feeling of respect for this science. Meanwhile, musicians encounter mathematics much more often than they themselves suspect, and, moreover, with such “terrible” things as logarithms. The famous physicist Eikhenwald recalled: “My high school friend loved to play the piano, but did not like mathematics. He even said with a touch of disdain that music and mathematics have nothing in common with each other. “It’s true that Pythagoras found some relationships between sound vibrations, but it was precisely the Pythagorean scale that turned out to be unacceptable for our music.” Imagine how unpleasantly my friend was amazed when I proved to him that when playing the keys of his piano, he was, strictly speaking, playing logarithms...” And indeed, the steps of the 12-sound scale of frequencies of sound vibrations are logarithms, the bases of which are equal to two. A logarithmic spiral is a plane curve described by a point moving in a straight line, which rotates about one of its points O (the pole of the logarithmic spiral) so that the logarithm of the distance of the moving point from the pole changes in proportion to the angle of rotation; a logarithmic spiral intersects at a constant angle all lines emanating from a pole. Sea animal shells can only grow in one direction. In order not to stretch too long, they have to twist, and each subsequent turn is similar to the previous one. And such growth can only occur in a logarithmic spiral. Therefore, the shells of many mollusks, snails, as well as the horns of mammals such as argali (mountain goats), are twisted in a logarithmic spiral. We can say that this spiral is a mathematical symbol for the relationship of the growth form. The great German poet Johann Wolfgang Goethe even considered it a mathematical symbol of life and spiritual development. Not only shells have outlines expressed by a logarithmic spiral. In a sunflower, the seeds are arranged in arcs, also close to a logarithmic spiral. One of the most common spiders, epeira, when weaving a web, twists the threads around the center in a logarithmic spiral. Many galaxies are also twisted in logarithmic spirals, in particular the galaxy to which the Solar System belongs.
