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.

Lesson Objectives:

1. Development of skills to systematize and generalize the properties of logarithms; apply them when simplifying expressions.
2. Development of conscious perception of educational material, visual memory, mathematical speech of students, to form skills of self-learning, self-organization and self-esteem, to promote the development of creative activity of students.
3. Fostering cognitive activity, instilling in students love and respect for the subject, teaching them to see in it not only rigor and complexity, but also logic, simplicity and beauty.

Equipment:

1. Interactive whiteboard (StarBoard Software)
2. Computers
3. Presentation 1“Logarithms. Properties of logarithms"
4. Presentation 2"Logarithms and Music"
5. Technological lesson map

Lesson type: a lesson in generalizing and systematizing knowledge. (Preparation for exams)

During the classes

I. Org. moment

1. Motivation

Dear Guys! I hope that this lesson will be interesting and of great benefit to everyone. I really want those who are still indifferent to the queen of all sciences to leave our lesson with a deep conviction: Mathematics is an interesting subject. The epigraph of the lesson will be the words of Aristotle: “It is better to do a small part of a task perfectly than to do ten times more poorly.”

(Slide 1. Interactive whiteboard or presentation 1). How do you understand these words?

2. Statement of the problem.

On slide 2 you see Portrait of Pythagoras, notes and logarithms. What do they have in common? (Slide 2 on the interactive whiteboard or slide 2-3 of the presentation 1).

3. Logarithms in music

(Slide 3 on the interactive whiteboard or slide 4 of the presentation 1).

In his poem “Physicists and Lyricists,” the poet Boris Slutsky wrote.

Even the fine arts feed on it.

Isn't the musical scale a set of advanced logarithms?

(Student message - presentation attached)

4. Lesson topic(Slide 4 on the interactive whiteboard or slide 5 of the presentation 1). The class is divided into three groups, each student has a technological map.

II. Repetition

1 group	2nd group	3 group
1. Repetition of theory
Insert missing words: Logarithm of a numberb By………………………. and is called …………….. the degree to which you need ……………. base a to get the numberb . build, base, indicator	In the technological map of the lesson - Task 1 Collect the definition of logarithm on the computer	In the technological map of the lesson - Task 1 Write down the definition of logarithm in mathematical language.
2. Self-test (Slide 5 on the interactive whiteboard or slide 7 of presentation 1)
3. Repetition of the properties of the logarithm (Slide 6-7 on the interactive whiteboard or slide 8-9 of presentation 1)
Task 2. Use arrows to connect the formulas on your computer.	Task 2. In the lesson flow chart, use arrows to connect the formulas	Task 2. Complete the formulas in the lesson plan
4. Peer review (Slide 8 on the interactive whiteboard or slide 10 of presentation 1)
5. Applying properties
a) Orally (Slide 9-10 on the interactive whiteboard or slide 11-12 of presentation 1) Calculate and match the answers
b) Find mistakes (Slide 11 on the interactive whiteboard or slide 13 of presentation 1)
c) Work in groups
Work at the board. Calculate	Executing a test in a routing Calculate:	Performing a test on a computer
6. Repetition of properties (Slide 12 on the interactive whiteboard or slide 14 of presentation 1)
7. Applying properties (Slide 13 on the interactive whiteboard or slide 15 of presentation 1)
Calculate:
8. Sophistry (Slide 14 on the interactive whiteboard or slide 16 of the presentation 1)
(from the Greek sophisma - trick, invention, puzzle), reasoning that seems correct, but contains a hidden logical error and serves to give the appearance of truth to a false statement. Usually sophistry substantiates some deliberate absurdity, absurdity or paradoxical statement that contradicts generally accepted ideas
8. Logarithmic sophism 2>3.(Slide 15 on the interactive whiteboard or slide 17 of the presentation 1)
Let's start with inequality, which is undoubtedly true. Then comes the transformation , also beyond doubt. A larger value corresponds to a larger logarithm, which means , i.e. . After reduction by , we have 2>3.

III. Homework

In the exam folder

Topic: “Properties of logarithms”

• 1st group - 1 option
• 2nd group - 2nd option
• 3rd group - 3rd option

IV. Lesson summary

(Slide 16 on the interactive whiteboard or slide 18 of the presentation 1)

“Music can uplift or soothe the soul,
Painting is pleasing to the eye,
Poetry is to awaken feelings,
Philosophy is to satisfy the needs of the mind,
Engineering is to improve the material side of people's lives,
A mathematics can achieve all these goals.”
So said the American mathematician Maurice Kline.

Thanks for the work!

A. Diesterweg

DEVELOPMENT AND EDUCATION CAN NOT BE GIVEN OR COMMUNICATE TO ANY PERSON. ANYONE WHO WISHES TO JOIN THEM MUST ACHIEVE THIS BY OWN ACTIVITY, OWN STRENGTH, OWN TENSION .

Determine the topic of the lesson by solving equations

• 2 x = ; 3 x = ; 5 x = 1/125; 2 x = 1/4; 2 x = 4; 3 x = 81; 7 x = 1/7; 3 x = 1/81

Logarithm and its properties

John Napier, inventor of logarithms

In 1590, he came up with the idea of ​​logarithmic calculations and compiled the first tables of logarithms, publishing the work “Description of Amazing Tables of Logarithms.” This work contained a definition of logarithms and an explanation of their properties. Invented the slide rule, a calculating tool that used Napier tables to simplify calculations.

Logarithmic ruler

Nowadays, with the advent of compact calculators and computers, the need to use tables

Logarithms and slide rules are no longer needed.

• The logarithm of the number a 0 to the base a 0 and a 1 is the exponent to which the number a must be raised to obtain the number b.
• - logarithm with an arbitrary base.
• For example: a) log 3 81 = 4, since 3 4 = 81; b) log 5 125 = 3, since 5 3 = 125; c) log 0.5 16 = -4, since (0.5) -4 = 16;

Application of logarithm: Banking calculations, geography, calculations in production, biology, chemistry, physics, astronomy, psychology, sociology, music.

Logarithmic spiral in nature

Nautilus shell

Arrangement of seeds on a sunflower

Properties of logarithms

• log a 1 = 0.
• log a a = 1.
• log a xy = log a x + log a y.
• log a x ∕ y = log a x - log a y.
• log a x p = p log a x
• log a р x = 1 ∕ р log a x

• If the base of the logarithm is 10, then the logarithm is called decimal:

• If the base of the logarithm is e 2.7, then the logarithm is called natural:

• 1. Find the base 4 logarithm of 64.

Solution: log 4 64 = 3, since 4 3 = 64.

Answer: 3

• 2. Find the number x, if log 5 x = 2

Solution: log 5 x = 2, x= 5 2 (by definition of logarithm), x = 25.

Answer : 25.

• 3. Calculate: log 3 1/ 81 = x ,

Solution: log 3 1/ 81 = x , 3 x = 1/ 81, x = – 4.

Answer: – 4.

• 1. Calculate: log 6 12 + log 6 3

Solution:

log 6 12 +log 6 3 = log 6 (12*3) = log 6 36 = log 6 6 2 = 2

Answer : 2.

• 2. Calculate: log 5 250 – log 5 2.

Solution:

log 5 250 – log 5 2 = log 5 (250/2) = log 5 125 = 3

Answer : 3.

• 3. Calculate:

Solution :

Answer: 8.

Slide 2

Lesson objectives:

Educational: Review the definition of logarithm; get acquainted with the properties of logarithms; learn to apply the properties of logarithms when solving exercises.

Slide 3

Definition of logarithm

The logarithm of a positive number b to base a, where a > 0 and a ≠ 1, is the exponent to which the number a must be raised to obtain the number b. Basic logarithmic identity alogab=b (where a>0, a≠1, b>0)

Slide 4

History of logarithms

The word logarithm comes from two Greek words and it is translated as a ratio of numbers. During the sixteenth century. The volume of work associated with carrying out approximate calculations in the course of solving various problems, and primarily the problems of astronomy, which has direct practical application (in determining the position of ships by the stars and the Sun), has sharply increased. The greatest problems arose when performing multiplication and division operations. Attempts to partially simplify these operations by reducing them to addition did not bring much success.

Slide 5

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. Let us add that 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. 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 included the values ​​of logarithms of sines, cosines and tangents for angles from 0 to 900 in steps of 1 minute. Burgi prepared his tables of logarithms of numbers, but they were published in 1620, after the publication of Napier's tables, and therefore went unnoticed. Napier John (1550-1617)

Slide 6

The invention of logarithms, by reducing the work of the astronomer, extended his life. P. S. Laplace Therefore, the discovery of logarithms, which reduces the multiplication and division of numbers to the addition and subtraction of their logarithms, lengthened, according to Laplace, the life of calculators.

Slide 7

Properties of degree

ax ay = ax +y = ax –y (x)y = ax y

Slide 8

Calculate: