Definition of the derivative. Middle line. Examination of the function for monotonicity. Works: Consolidation of the studied material. Calculate approximate using differential. The smallest values \u200b\u200bof the functions. Derivative and its application in algebra, geometry. The function in question. A task. Inequality. Signs of increasing and decreasing functions. Dot. Definition. Finding the differential. Proof of inequalities.

"" Integral "Grade 11" - As you defeated lay the usual number on the page. Integral in literature. A definite integral, I started dreaming about you at night. Compose a phrase. What happiness I have learned in choosing the primitive. Zamyatin Evgeny Ivanovich (1884-1937). Find antiderivatives for functions. Epigraph. The novel "We" (1920). A number of substitutions and substitutions led to the solution of the problem. Illustration for the novel "We". Integral. Integral group. An algebra lesson and the beginnings of analysis.

"Application of logarithms" - Since the time of the ancient Greek astronomer Hipparchus (II century BC) the concept of "magnitude" has been used. As we see, logarithms invade the field of psychology. From the table we find the stellar magnitude of Capella (m1 \u003d + 0.2t) and Deneb (m2 \u003d + 1.3t). Volume unit. Stars, noise and logarithms. The harmful effect of industrial noise on workers' health and labor production. Topic: "LOGARITHMS IN ASTRONOMY". Napier (1550 - 1617) and the Swiss I. Burghi (1552 - 1632).

"Functions" algebra "- Calculate. Let's make a table. The study of functions and the construction of their graphs. Integral concept. The function F is called the antiderivative for the function f. Curved trapezoid area. A function is an antiderivative for a function. Let us calculate the area S of the curved trapezoid. "Integral from a to b eff from x de x". The method of intervals. Let's find the points of intersection of the graph with Ox (y \u003d 0). Differentiation rules. Let's find the largest and smallest values \u200b\u200bof the function on the segment.

"Examples of logarithmic inequalities" - Getting ready for the exam! Which of the functions are increasing and which are decreasing? Lesson summary. Find the right solution. Increasing. Algebra grade 11. Task: solve the logarithmic inequalities proposed in the tasks of the exam-2010. Good luck on the exam! Cluster to fill during the lesson: Lesson objectives: Find the scope of the function. Between the numbers m and n put a\u003e or<.(m, n > 0). Graphs of logarithmic functions.

"Geometric meaning of the derivative of a function" - The value of the derivative of a function. Algorithm for drawing up a tangent equation. The geometric meaning of the derivative. Equation of a straight line with a slope. Tangent equations. Make a pair. Secant. Lesson vocabulary. I did it. Correct mathematical idea. Calculation results. The limiting position of the secant. Definition. Find the slope. Write the equation of the tangent line to the graph of the function.

Lesson objectives:

1. Practicing skills to systematize, generalize the properties of logarithms; apply them when simplifying expressions.
2. The development of conscious perception of educational material, visual memory, mathematical speech of students, to form the skills of self-study, self-organization and self-esteem, to promote the development of creative activity of students.
3. Fostering cognitive activity, fostering love and respect for the subject in students, teaching them to see in it not only rigor, complexity, but also consistency, 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 generalization and systematization of knowledge. (Exam preparation)

During the classes

I. Org. moment

1. Motivation

Dear Guys! I hope that this lesson will be interesting, with great benefit for 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 "Better to do a small part of the job perfectly, than to do ten times more badly."

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

2. Statement of the problem.

On slide 2 you can see the Portrait of Pythagoras, notes and logarithms. What unites them? (Slide 2 on the whiteboard or slide 2-3 of the presentation 1).

3. Logarithms in music

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

In his poem "Physics and Lyrics" poet Boris Slutsky wrote.

Even the fine arts feed on it.

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

(Student message - presentation attached)

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

II. Reiteration

1st group	2nd group	Group 3
1. Repetition of theory
Insert missing words: Logarithm of the numberb by………………………. but 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 the logarithm on a computer	In the technological map of the lesson - Task 1 Write the definition of a logarithm in mathematical language.
2. Self-test (Slide 5 on the 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 formulas on the computer	Task 2. In the flowchart of the lesson, use arrows to connect the formulas	Task 2. Complete the formulas in the lesson flowchart
4. Peer review (Slide 8 on the whiteboard or Slide 10 of presentation 1)
5. Applying properties
a) Orally (Slide 9-10 on the whiteboard or slide 11-12 of presentation 1) Calculate and match answers
b) Find mistakes (Slide 11 on the whiteboard or slide 13 of presentation 1)
c) Working in groups
Work at the blackboard. Calculate	Test execution in a routing Calculate:	Running a test on a computer
6. Repetition of properties (Slide 12 on the whiteboard or slide 14 of presentation 1)
7. Applying Properties (Slide 13 on the Whiteboard or Slide 15 of Presentation 1)
Calculate:
8. Sophism (Slide 14 on the whiteboard or slide 16 of the presentation 1)
(from the Greek sophisma - trick, invention, puzzle), reasoning that seems to be correct, but contains a hidden logical error and serves to make a false statement appear to be true. Usually sophism substantiates some deliberate absurdity, absurdity or paradoxical statement that contradicts generally accepted ideas
8. Logarithmic sophism 2\u003e 3.(Slide 15 on the whiteboard or slide 17 of the presentation 1)
Let's start with inequality, which is undeniably true. Then comes the transformation , which is also beyond doubt. The larger value corresponds to the larger logarithm, which means that , i.e. . After canceling by, we have 2\u003e 3.

III. Homework

In the exam folder

Topic: "Properties of logarithms"

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

IV. Lesson summary

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

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

Thanks for your work!

Lesson topic:

Logarithms and their properties.

Esmaganbetov K.S. Mathematic teacher.

The purpose of the lesson:

1. Development of skills to systematize, 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 the skills of self-study, self-organization and self-esteem, to promote the development of creative activity of students.

3. Education of cognitive activity, to foster in students love and respect for the subject, to teach to see in it not only severity, complexity, but also logic, simplicity and beauty.

I. Brainstorming:

1) What is an antiderivative?

2) What kinds of integrals do you know?

3) What is the difference between a definite integral and an indefinite one?

4) What equations are called irrational?

5) How many rules are there for finding antiderivatives?

Questions:

Working in groups

• Define the topic of the lesson using an anagram:
• YMFIRAOL AND KHI AVTSYOVS
• Evaluation criteria for guessing the anagram (for the correct answer-1 point, for the wrong answer-0 point)

Logarithms and their properties

• Logarithm base a of a positive number b, where a\u003e 0, a ≠ 1, is the exponent to which the number a must be raised to obtain b.
• Basic logarithmic identity:
alogab \u003d b,where b\u003e 0, a\u003e 0
• If the base of the logarithm is 10, then this logarithm is called decimal.
• If the base of the logarithm is equal to the number e, then such a logarithm is called natural

Properties of logarithms

• The logarithm of the base itself is 1:
logaa \u003d 1
• The logarithm of one to any base is zero:
loga1 \u003d 0
• The logarithm of the product of two or more positive numbers is equal to the sum of the logarithms of the factors:
loga (bc) \u003d logab + logac
• The logarithm of the quotient of positive numbers is equal to the difference between the logarithms of the dividend and the divisor:
loga (b / s) \u003d logab - logac
• The logarithm of the power is equal to the product of the exponent by the logarithm of its base:
logabn \u003d n logab
• The formula for the transition from base b to base a:
Logax \u003d logbx / logba

Criteria for evaluating the technological map:

• Provide mathematical information clearly and logically-1 point;
• The student shows knowledge of mathematical symbols-1 point;

Calculate orally:

Evaluation criteria for oral computation

• for correct oral calculation-1 point
• for incorrect verbal calculation-0 points

Fizminutka

• Two halves

loga (x / y) loga x -loga y

Group work:

Task for the 1st group

Group work: Assignment to the 2nd group In the lesson flow chart, use arrows to connect the formulas

• logax + logay

Group work: Assignment to the 3rd group In the flow chart of the lesson, complete the formulas Mutual assessment Mutual assessment criteria

• for the correct finding of formulas-by 1 point group;
• For incorrectly finding formulas-0 points.

Individual written work on differentiated assignments

	log 26 - log 2 (6/32)
	log 3 5 - log 3 135
	2 log 27 - log 2 49
	log 93+ log 9243

Solution of Individual work on differentiated tasks

		lg (8 ∙ 125) \u003d lg 1000 \u003d 3
	log 26 - log 2 (6/32)	log 2 (6: (6/32)) \u003d log 232 \u003d 5
	log 3 5 - log 3 135	log 3 (5: 135) \u003d log 3 (1:27) \u003d -3
	2 log 27 - log 2 49	log 272 - log 249 \u003d log 2 (49:49) \u003d log 2 1 \u003d 0
	log 93+ log 9243	log 9 (3 ∙ 243) \u003d log 9729 \u003d 3

Evaluation Criteria for Individual Written Work

• for the correct solution of examples in full-5 points;
• For the correct spelling of mathematical symbols-1 point;

Development of criteria for evaluating work results:

• Evaluation criteria: for 20 points and above - score "5"
• for 16-19 points and above - score "4"
• for 9-15 points and above - grade "3"

Creation of clusters and their protection Criteria for evaluating clusters:

• For the correct creation of the cluster-1 point;
• For the elegance of the cluster design - 0.5 points;
• For good protection of the cluster-1 point

Reflection

• 1. What do I know about ____
• 2. What I want to know _____
• 3. What I learned ____
• 4. Assess your work in the lesson _____

Homework

1. Make a syncwine "Logarithms"

2. Assignment for the textbook: No. 241, No. 242

The logarithm is a fairly extensive topic in the algebra course for high school students, so knowing only its definition, a mathematical formula and being able to draw a graph is not enough. Throughout the history of the logarithmic formula, mathematicians from all over the world have derived a large number of dependencies and theorems, the knowledge of which will help students in their further work with this function.

The presentation "Properties of Logarithms" gives a broad understanding of this definition, and also allows you to familiarize yourself with all the most important consequences of this function.

The first part of the presentation briefly introduces the concept of a logarithm, and also demonstrates how to plot a graph based on it. This is followed by the definition to be learned, which is confirmed by the exclamation mark in the corner of the red frame.

After restoring knowledge on a previously studied topic, students are invited to familiarize themselves with three identical equations, which can be easily proved by any student who has to operate with concepts such as the degree of a number and the base of the degree.

The third part of the lesson is theoretical. Here, students are shown three theorems that are based on various mathematical operations with logarithms, including when working with fractions. Each theorem is highlighted with a blue box, below which is the mathematical proof.

After the theoretical part of the presentation, students have the opportunity to apply their new knowledge in practice, by considering the solution of one example.

The presentation ends with one more theorem, as well as three examples of solving problems based on the properties of logarithms. The last theorem proposed in the lesson does not require the ability to prove it in a regular school algebra course - it is enough for the student to memorize, understand and be able to apply it when solving thematic examples.

In contrast to the usual algebra course, which is offered by a school textbook, the presentation "Properties of logarithms" has a completely different, more convenient and effective structure that allows you to convey the required knowledge to the student as quickly and easily as possible. The presentation dilutes the theoretical part with practical examples that switch the student's attention to another activity, thereby not overloading his brain and giving the opportunity to take a break from the change in mental activity.

A quick understanding of the solutions to the proposed examples is facilitated by an interesting concept of information presentation, which is very difficult to find in an ordinary 11th grade algebra textbook. In the tasks proposed for consideration in the presentation, the most important data are highlighted in red or surrounded by a frame. This technique allows not only to quickly assimilate the most important information, but also teaches the student to independently search for the necessary material from the entire context.

The section of modern algebra "properties of logarithms" is one of the most important in the entire course, as it provides a foundation for further, in-depth study of mathematics, which is necessary for hundreds of modern professions, relating to various spheres of human life. It is for this reason that you should not ignore this topic, and if a student, for some reason, missed her studies at school, then the presentation of the "properties of logarithms" will help him catch up to the full, thanks to the easy and accessible presentation of the material in the lesson ...

The presentation of "properties of logarithms" is designed in such a way that it will be comfortable for both students and teachers to work with it: all information has a complete form on a separate page, so the lesson can not only be shown using various modern devices, but also simply printed if the school has no other capabilities.

JOHN NEPER (1550-1617)

Scottish mathematician -

inventor of logarithms.

In the 1590s came to the idea

logarithmic calculations

and made the first tables

logarithms, but its famous

the work “Description of amazing tables of logarithms” was published only in 1614.

He owns 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 due to the needs of computational practice.

• In those days, very cumbersome calculations had to be done to solve the problems of astronomy and navigation.

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

• The word "logarithm" is of Greek origin, which means the ratio of numbers

0, and ≠ 1 is called the exponent to which the number a must be raised to get b. "width \u003d" 640 "

Definition

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

Calculate:

log 2 16; log 2 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 the definition of the 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 \u003d" 640 "

At what values x there is a logarithm

Does not exist at any

what x

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

log a (bc) \u003d log a b + log a c

( b

c )

a log a (bc) =

a log a b

\u003d 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) \u003d log a b + log a c

Example:

log a

\u003d 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 \u003d a log a b

c \u003d a log a c

0; a ≠ 1; b 0; c 0. Example: 1 "width \u003d" 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

\u003d log a b - log a c,

a 0; a ≠ 1; b 0; c 0.

Example:

0; b 0; r R log a b r \u003d r log a b Example a log a b \u003d b 1,5 (a log a b) r \u003d b r a rlog a b \u003d b r "width \u003d" 640 "

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

log a b r \u003d r log a b

Example

a log a b \u003d b

(a log a b ) r \u003d b r

a rlog a b \u003d b r

Single base transition formula

logarithm to another, examples.