Calculation of logarithms examples with solution presentation. Presentation for the lesson "Logarithms - a whim of mathematicians, or a vital necessity"? Decimal and natural logarithms

Description:

On the topic of “logarithms,” this educational material comprehensively reveals the essence.

Conducting lessons using this methodological material Provides concepts 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 frequently and possible methods solutions.

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.

Slides:

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Date of material creation: May 07, 2013
Slides: 10 slides
Presentation file creation date: May 07, 2013
Presentation size: 22 KB
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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.

With the help of such slide rules, Soviet engineers performed calculations when designing buildings, structures, large industrial facilities, built in the USSR, new aircraft, cars, 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.”

3" Comedy begins with an inequality that is undeniably right. Then follows a transformation that is also not in doubt. A larger number corresponds to a larger logarithm, which means After reduction by lg we get: 23. What is the error in this reasoning? The error is that lg" width="640"

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 a 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 pH - 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% the pawnshop paid at that time.

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."

Carrying out this work, we made a discovery for ourselves 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

With

log

With

log

–

With

log

With

log

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 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.

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

0, a ≠1 is the exponent to which the number a must be raised to obtain b. "width="640"

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 .

3 X X X R Does not exist for any x " width="640"

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

0; a ≠ 1; b 0; c 0. Example: 1 " width="640"

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:

0; b 0; r R log a b r = r log a b Example a log a b =b 1.5 (a log a b) r =b r a rlog a b =b r " width="640"

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.

Radioactive decay The change in the mass of a radioactive substance occurs according to the formula N = N0 2-t/T, where N0 is the mass of the substance at time t, T is a certain constant, the meaning of which we will now find out. Let's calculate the value of N, at t=T. So, N(T)=N0*2^-1=N0/2. This means that after time T after the initial moment, the mass of the radioactive substance is halved. Therefore, the T number is called the half-life. The half-life of radium is 1600 years, uranium is 5 billion years, cesium is a year, iodine is a day. The law of radioactive decay is often written in the standard form N= No L- t \ T. The relationship between the constant T and the half-life is easy to find: L-t\T=2 –t\ T= -t\T ln2=T\ ln2~ 1.45 T.

Barometric formula Air pressure decreases with altitude (at constant temperature) in law. P=Po L- h\H, where p is the pressure at level h, H is a certain constant depending on temperature. For a temperature of 20, the value of H is ~7.7 kilometers. The sound insulation coefficient of walls is measured by the formula: D=A log Po\P, where Po is the sound pressure before absorption, P is the sound pressure passing through the wall, A is a certain constant, which in calculations is taken equal to 20 dB. If the sound insulation coefficient D is equal to, for example, 20 DB, then this means that Log Po\ P = 1 and Po = 10p, i.e. the wall reduces the sound pressure by 10 times (a wooden door has such sound insulation).

Tsiolkovsky's formula. This formula connecting the speed of the rocket V with its mass m is as follows: V=Vr ln m o \m, where Vr is the speed of the emitted gases, m o is the launch mass of the rocket. The gas outflow velocity V r- during fuel combustion is small (at present) it is less than or equal to 2 km/s. Logarithmic calculation is very slow, and in order to achieve escape velocity, it is necessary to make the ratio m o \m large, i.e. give almost the entire starting mass for fuel.

Definition of logarithm. Basic logarithmic identity. Decimal and natural logarithms.

MATH TEACHER:

Alexandrina Lyudmila Vladimirovna

GBPOU "Muravlenkovsky College"

Yamalo-Nenets Autonomous Okrug, Muravlenko

The purpose of the lesson:

- Define the logarithm and its properties, the basic logarithmic identity

- Show the usefulness of using logarithms;

- Teach to see the familiar in the unfamiliar, develop an interest in the history of mathematics and its applications.

Find the positive root of the equation

x 2 = 9 answer: x=3

x 3 = 8 answer x = 2

x 4 = 81 answer: x=3

Solve the equation

2 x =8 answer: x=3

3 x =27 answer: x=3

5 x =7 answer: ?

0 and a 1 is the exponent to which the number a must be raised to obtain the number b. = x Logarithm with an arbitrary base." width="640"

Definition of logarithm

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

Logarithm with an arbitrary base.

Logarithms with base 10 are called decimal.

Designation:Lg

For example: Lg100=2

Logarithms with base e = 2.718... are called natural.

Designation: Ln

Basics logarithmic identity

The act of finding the logarithm of a number is called logarithm

Calculate

loq 3 27=

loq 5 125=

loq 2 2=

loq 8 1=

loq 2 16=

loq 3 9=

3 loq 3 18 =

loq 0,5 0,25=

loq 2 x=3

7 loq 7 3 =

Calculate

loq 4 1=

loq 13 13=

loq 3 x=2

6 loq 6 12 =

loq 4 x=2

loq 2 x=5

loq 13 13=

loq 3 x=2

5 loq 5 12 =

loq 9 1=

Calculate it yourself

loq 3 3=

loq 2 16=

loq 2 x=3

3 loq 3 18 =

loq 2 2=

loq 2 64=

loq 15 15=

loq 3 x=2

4 loq 4 12 =

loq 9 1=

Logarithmic warm-up "A little history."

Logarithm - from Greek. λόγος - “word”, “attitude” and ἀριθμός - “number”, “indicator”

The applications of exponential and logarithmic functions in various fields of science and technology are truly limitless, and logarithms were invented to facilitate calculations. Four 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 thereby, as the famous French scientist Laplace said, “lengthened the life of calculators.”

Logarithmic warm-up "A little history."

In parallel with Napier on the compilation

logarithm tables worked differently

math lover - Jost Bürgi.

He was a Swiss watchmaker and

master of astronomical instruments.

Bürgi compiled tables of logarithms

earlier, but only in 1620 he published his

book "Tables of Arithmetic and

geometric progression with detailed

instructions on how to use them

all kinds of calculations."

Logarithmic warm-up "A little history."

In 1623, i.e. just 9 years after publication

first tables, English mathematician Edmund

Gunther the first logarithmic was invented

a ruler that has become a working tool for many

generations until the advent of computers.

Logarithmic spiral “The amazing is nearby »

A spiral is a flat curved line that repeatedly circles one of the points on the plane, which is called the pole of the spiral.