Logarithm of a number. Properties of logarithms.
GBOU TsO No. 173 Popova L.A.
Definition of logarithm
The logarithm of a positive number b to base a, a>0,a≠1, is the exponent to which the number a must be raised to obtain the number b.
A decimal logarithm is a logarithm to the base. Symbol:
Natural logarithm is a logarithm to the base e (e is an irrational number, the approximate value of which is: e = 2.7. Designation:
Basic logarithmic identity
, Where
Properties of logarithms
Logarithm of unit
Logarithm of the product of positive numbers
Logarithm of the quotient of positive numbers
Logarithm of powers of positive numbers
Formula for moving from one logarithm base to another
Consequences
Calculate:
log 464=
52 5log53=
lg1=
log0,1 =
log381=
log77 =
log1/216=
log12√ 144
log3√100=
log1/31/81=
log1/21/32=
log5125
log23√2=
log1/749
lg0.001 =
log2 log 381=
lg10000=
log2 log 5625=
Verbal counting
Formula for converting to decimal and natural logarithms
Replace this logarithm with a base 3 logarithm:
1.
2.
.3
4.
5.

On the topic: methodological developments, presentations and notes

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JOHN NAPER (1550-1617)

Scottish mathematician

inventor of logarithms.

In the 1590s he came up with the idea

logarithmic calculations

and compiled the first tables

logarithms, but its famous

The work “Description of Amazing Tables of Logarithms” was published only in 1614.

He is responsible for the definition of logarithms, an explanation of their properties, tables of logarithms, sines, cosines, tangents and applications of logarithms in spherical trigonometry.

From the history of logarithms

• Logarithms appeared 350 years ago in connection with the needs of computing practice.

• In those days, very cumbersome calculations had to be made to solve problems in astronomy and navigation.

• The famous astronomer Johannes Kepler was the first to introduce the logarithm sign – log in 1624. He used logarithms to find the orbit of Mars.

• The word “logarithm” is of Greek origin, which means ratio of numbers

Definition

The logarithm of a positive number b to base a, where a0, a ≠1 is the exponent to which the number a must be raised to obtain b.

Calculate:

log 2 16; log2 64; log 2 2;

log 2 1 ; log 2 (1/2); log 2 (1/8);

log 3 27; log 3 81; log 3 3;

log 3 1; log 3 (1/9); log 3 (1/3);

log 1/2 1/32; log 1/2 4; log 0.5 0.125;

Log 0.5 (1/2); log 0.5 1; log 1/2 2.

Basic logarithmic identity

By definition of logarithm

Calculate:

3 log 3 18 ; 3 5log 3 2 ;

5 log 5 16 ; 0.3 2log 0.3 6 ;

10 log 10 2 ; (1/4) log (1/4) 6 ;

8 log 2 5 ; 9 log 3 12 .

At what values X there is a logarithm

Doesn't exist at all

which X

1. The logarithm of the product of positive numbers is equal to the sum of the logarithms of the factors.

log a (bc) = log a b + log a c

( b

c )

a log a (bc) =

a log a b

= a log a b + log a c

a log a c

a log a b

a log a c

1. The logarithm of the product of positive numbers is equal to the sum of the logarithms of the factors. log a (bc) = log a b + log a c

Example:

log a

=log a b-log a c

= a log a b - log a c

a log a b

a log a

a log a c

b = a log a b

c = a log a c

2. The logarithm of the quotient of two positive numbers is equal to the difference between the logarithms of the dividend and the divisor.

log a

=log a b–log a c,

a 0; a ≠ 1; b 0; c 0.

Example:

3. The logarithm of a power with a positive base is equal to the exponent times the logarithm of the base

log a b r = r log a b

Example

a log a b =b

(a log a b ) r =b r

a rlog a b =b r

Formula for moving from one base

logarithm to another, examples.

Practical application of logarithms

Logarithmic functions are extremely common in both mathematics and the natural sciences. A number of natural phenomena help to describe the logarithmic dependence. In other words, mathematicians, when trying to create a mathematical model of a particular phenomenon, quite often turn to the logarithmic function.

One of the most obvious examples is the logarithmic spiral. The spiral unfolds to infinity in one direction, and, on the contrary, twists around the pole, tending to it, but not reaching it.

Sea animal shells can only grow in one direction. In order not to stretch too much in length, they have to curl, and the growth occurs in such a way that the resemblance of a shell with its original shape is preserved. And such growth can only occur in a logarithmic spiral. Biology

One of the most common spiders, epeira, when weaving a web, twists the threads around the center in logarithmic spirals. Biology

The horns of mammals such as mountain goats are twisted in a logarithmic spiral. In a sunflower, the seeds are arranged in arcs close to a logarithmic spiral. Biology

Mechanics and physics The Boltzmann principle in statistical thermodynamics is one of the most important functions of the state of a thermodynamic system, characterizing the degree of its chaos. The Tsiolkovsky formula is used to calculate the speed of a rocket.

The unit of sound volume is “bel”, practically its tenth, “decibel”. A loudness difference of 1 bel corresponds to a noise intensity ratio of 10. This means that the loudness of the noise, expressed in bels, is equal to the decimal logarithm of its physical strength.

Chemistry Hydrogen index, "pH", is a measure of the activity of hydrogen ions in a solution, quantifying its acidity, calculated as the negative decimal logarithm of the concentration of hydrogen ions, expressed in moles per liter

Astronomy Many Galaxies are twisted in logarithmic spirals, in particular the Galaxy to which the Solar System belongs.

Astronomers classify stars according to degrees of apparent brightness into luminaries of the first magnitude, second magnitude, third, etc. It is easy to understand that the “magnitude” of a star is nothing more than the logarithm of its physical brightness. When estimating the apparent brightness of stars, the astronomer operates with a table of logarithms compiled at a base of 2.5.

Music The “steps” of the tempered chromatic scale are not spaced at equal distances either in relation to the vibration numbers or in relation to the wavelengths of the corresponding sounds, but represent logarithms of these values. Hence we see that the numbers of the piano keys are logarithms of the vibration numbers of the corresponding sounds.

Geography Richter proposed to estimate the strength of an earthquake (at its epicenter) using the decimal logarithm of the displacement (in micrometers) of the needle of a standard Wood-Anderson seismograph located at a distance of no more than 600 km from the epicenter.

Psychology The Weber-Fechner law is an empirical psychophysiological law, which states that the intensity of sensation is proportional to the logarithm of the intensity of the stimulus.

Psychology Fitts's Law is a general law that relates the time of movement to the accuracy of movement and the distance of movement: the further or more accurately a movement is performed, the more correction is necessary for its execution, and accordingly, the more time is required to make this correction

Psychology The time to make a decision when given a choice can be estimated using Hicks' law.

Computer science Used to calculate the basic unit - the bit. A bit is the binary logarithm of the probability of equally probable events or the sum of the products of the probability and the binary logarithm of the probability for equally probable events

Sources of information: http://ru.wikipedia.org/wiki/ http://s_2_petrop.ven.edu54.ru/p89aa1.html http://images.yandex.ru/?uinfo=ww-1341-wh-591 -fw-1274-fh-448-pd-1

Logarithms - a whim of mathematicians or a vital necessity?

Logarithms are rhymes

Like words in music.

They make calculations easier -

No more difficult than twice two.

L. Nesterova

Meeting of the Academic Council

• We will systematize and expand knowledge on the topic “Logarithms”;
• Let's consider the practical and theoretical applications of logarithms;
• Let's solve logarithms from Unified State Examination tasks;
• And just relax with the logarithms.

Meeting plan:

Greeting speech

Chairman of the Academic Council

Introduction to the world of logarithms from a mathematical perspective

Black and white opposition

History of the development of logarithms. Is logarithm fun?!

Black and white opposition

Is logarithm an ordinary mathematical concept or something more?

Black and white opposition

Logarithm in Unified State Examination tasks.

Mini competitions

Reflection

Mathematicians - theorists

Historians

Divide into groups

Scientists

Mathematicians - practitioners

Group "Theoretic Mathematicians"

Logarithms in mathematics

Determine the need to study logarithms in mathematics

Word "logarithm" comes from Greek words  - number And  - attitude . Translated as “relations of numbers”, one of which is a member of an arithmetic progression, and the other of a geometric one.

Dictionary of the Russian language by S. I. Ozhegova

Logarithm- in mathematics: an exponent to which a number, called the base, must be raised to obtain a given number.

Explanatory Dictionary of the Living Great Russian Language" by V. Dal

Logarithm. If under a series of numbers of a geometric progression (ladder) we put a series of corresponding numbers of an arithmetic progression, then each of the latter will be the logarithm of its friend, in the first order; In this way, multiplication is turned into addition, division into subtraction, which makes calculations easier.

“Realizing that in mathematics there is nothing more boring and tedious than multiplication, division, square and cube roots, and that these operations are a waste of time and an inexhaustible source of subtle errors, I decided to find a simple and reliable means to get rid of them "

John Napier, "The Canon of Logarithms"

John Napier ( 1550-1617)

Henry Briggs (1561-1631)

Briggs logarithm- the same as the decimal logarithm.

Named after G. Briggs

Decimal logarithm- logarithm to base 10. The decimal logarithm of the number a 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.

There is nothing in the world but Beauty.

There is nothing in Beauty except Form.

There is nothing in Form but proportions.

There is nothing in proportions except Number.

Pythagoras

"Golden" logarithms are logarithms with a base equal to the number

Ф (1, 6180339) are described by the formula

log F M = P

Three bases of logarithms:

10,000 ; 3,838 ; 2,71 .

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.

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.

Group "Historians"

The history of logarithms

Establish a picture of the origin of the concept of “logarithm”

The project involves the collection and analysis of data, their presentation in a clear visual form and is aimed at developing an understanding of the meaningful meaning of the term “logarithm”

By whom and when were logarithms introduced?

The invention of logarithms, while reducing the astronomer's work, extended his life.

P. S. Laplace

Archimedes (III century BC) - ancient Greek physicist, mechanic and engineer from Syracuse.

The work continued in the 16th century by the Scottish Baron Napier

Napier John (1550-1617) - 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,” which sets out this doctrine, was published in 1614.

This work contained a definition of logarithms, an explanation of their properties, tables of logarithms of sines, cosines, tangents, and applications of logarithms in spherical trigonometry.

Napier went down in history as the inventor of a remarkable computing tool - the table of logarithms. This discovery caused a gigantic relief in the work of the calculator.

Named after John Napier:

• Named after John Napier:

• crater on the Moon;

• asteroid 7096 Napier;
• a logarithmic dimensionless unit that measures the ratio of two quantities;
• University of Edinburgh

Slide Rule - Calculation Tool

In 1623, the English mathematician D. Gunther invented the first slide rule, which became a working tool for many generations.

The principle of operation of a slide rule is based on the fact that the multiplication and division of numbers is replaced by the addition and subtraction of their logarithms, respectively.

Using such slide rules, Soviet engineers performed calculations when designing buildings, structures, large industrial facilities being built in the USSR, new aircraft, cars, and ships. It was used by accountants and specialists who would now be called managers. Once upon a time, slide rules made life much easier for students.

Nowadays, inexorable progress has consigned slide rules to oblivion and left them a place only on a museum shelf.

Celebrities and the spiral

The logarithmic spiral was first mentioned in a letter by the French mathematician Rene Descartes in 1638.

The great German poet Johann Wolfgang Goethe considered the logarithmic spiral to be a mathematical symbol of life.

The logarithmic spiral so impressed the mathematician Jacob Bernoulli that he bequeathed to carve its image on his tombstone along with the inscription in Latin “Changed, I am reborn as before.”

LOGARITHMIC "COMEDY 2 3"

Comedy begins with inequality undeniably correct.

Then comes the transformation also not inspiring doubt.

A larger number corresponds to a larger logarithm, which means

After reduction by lg we get: 23.

What is wrong with this reasoning?

The error is that lg

Group "Science Scientists"

Are logarithms ordinary mathematical concepts or something more?!

In what sciences are logarithms used?

How are many real objects in astronomy, biology, physics, chemistry and other natural sciences related to logarithms?

“The invention of logarithms, while reducing the work of the astronomer, extended his life.”

In the 2nd century BC. Hipparchus divided the stars into 6 groups. The brightest stars are 1st magnitude, the faintest are 6th magnitude.

It was established that the star was the 1st led. brighter than the star 6th led. exactly 6 times.

• star 1 led. brighter than the sound 2 vel. in 2.512;
• star 1 led. brighter than the sound 3 vel. B 2.512 2;

The scope of application of logarithms is very diverse: mathematics, literature, biology, psychology, agriculture, music, astronomy, physics

So astrologers, when assessing the apparent brightness of stars, operate with a table of logarithms compiled

at base 2.512.

The “magnitude” of a star is nothing more than the logarithm of its physical brightness.

• pH value - it is a measure of the activity of hydrogen ions in a solution, quantifying its acidity, calculated as the negative decimal logarithm of the concentration of hydrogen ions, expressed in moles per liter:
• pH = -lg

Logarithmic spiral in technology

And we see this spiral everywhere: For example, rotating the knives in the mechanism. We will find it in the bend of the pipe - The turbines will then serve as much as possible!

The magnitude of the sensation is proportional to the logarithm of the magnitude of stimulation

The piano key numbers are logarithms of the vibration numbers of the corresponding sounds.

• One of the most common spiders, EPEIRA, when weaving a web, twists the threads around the center in a logarithmic spiral

• Many galaxies are twisted in logarithmic spirals, in particular, the galaxy to which the Solar System belongs

Spiral Galaxy Whirlpool

An interesting problem taken from the book “The Golovlevs” by Saltykov-Shchedrin:

“ Porfiry Vladimirovich sits in his office, writing sheets of paper with numbers. This time he is occupied with the question: How much money would he have if his mother had not appropriated the 100 rubles given to him at birth by his grandfather for a tooth, but put it in a pawnshop in the name of young Porfiry? It turns out, however, not much: only 800 rubles?

Assuming that Porfiry was 50 years old at the time of calculation, and making the assumption that he made the calculations correctly (an unlikely assumption, since Golovlev hardly knew logarithms and knew how to calculate compound interest), it is necessary to establish how much% he paid at that time pawnshop.

However, at the beginning XXI century slide rules have been reborn in wristwatches hours. The fact is that, following the fashion, manufacturers of expensive and prestigious watch brands switched from electronic chronometers with LCD screens to dial ones and, accordingly, there was not enough space for a built-in calculator. However, the demand for chronometers with a built-in computing device among fashion-conscious people forced watch manufacturers to release models with a built-in slide rule made in the form of rotating rings with scales around the dial.

The applications of the logarithmic function and logarithms in various fields of science and technology are truly limitless.

The multiple uses of the function inspired the English poet E. Brill to write an ode about logarithms.

There were poets who did not devote entire odes to logarithms, but mentioned them in their poems. The famous poet Boris Slutsky wrote in his acclaimed poem “Physicists and Lyricists”:

“That’s why, like foam,

Our rhymes fall

And greatness sedately

Retreats into logarithms."

While carrying out this work, we made the discovery that logarithms and the logarithmic function helped people follow the path of technical progress and explain many of the secrets of nature and human sensations. Perhaps humanity is on the verge of new revolutionary discoveries, and the “queen of sciences” - mathematics - will help us in this!

Group "Practical Mathematicians"

The purpose of our work:

show solutions to examples taken from Unified State Examination tasks.

We have set ourselves the task:

show that knowledge about logarithms is also necessary for the Unified State Examination in mathematics.

Only knowing all the properties of the logarithm can you learn to solve examples

log

a

=

log

+

b

With

log

With

b

log

=

a

a

a

a

log

b

b

b

b

a

=

log

–

b

a

a

With

log

a

=

log

With

a

a

a

b

log

b

log

log

c

b

log

b

=

=

b

a

a

a

a

log

log

b

c

log

b

log

a

=

r

b

a

log

b

r

log

b

=

a

a

b

log

log

b

=

r

r

a

r

a

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.

Mini competitions

Competition No. 1

What is the name of the mathematician who continued Napier's work on creating tables of logarithms?

Answer Key:

Answer: Briggs

Competition No. 2

Indicate the geographic coordinates of the island of Jan Mayen, where Napier, the creator of logarithms, lived.

Geographical coordinates:

X°00′ north latitude, at°00′ west longitude.

To find x and y, solve the equations:

71°00′ north latitude, 8°00′ west longitude

Black box

Here lies the result of the work of many scientists. What is located here was used in educational institutions and engineering calculations until the end of the last century.

Here lies what the English mathematician William Oughtred came up with back in the 20s of the 17th century.

FOUNDATION

h a s t n o o

P o x a t e l

tenth

l o g a r i m i o n

Favorite number

Now please take your pens and write down your favorite number.

Multiply this number by 9. Multiply the resulting number by 12345679.

If you did it correctly, you will get a bouquet of your favorite numbers. Now add 9 zeros to the right of the resulting number. May there be so many happy days in your life.

Description:

Conducting lessons using this teaching material will teach the concept to students using visual images and logically structured material. Using detailed solved tasks as examples along with illustrations of graphs will help each student understand. The tasks are typical and divided into groups. This helps to systematically study the submitted material and allows you to see the possible types of tasks that occur most often and possible methods of solution.

The parts of the presentation are:

• How to determine logarithm based on base.
• It talks about eight properties of the logarithm, which are basic.
• Describes natural and decimal logarithms.
• Attention is paid to the logarithmic function itself and its properties.
• The technique for solving equations, systems of equations, and also inequalities is practically illustrated.

The presentation will be convenient to use not only as a timely educational source of information in the lesson, but also during the re-recovery of the material in the student’s memory.

Category:

Slides:

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