The logarithm is a rather extensive topic in an algebra course for high school students, so knowing only its definition, 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 deduced 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 an extensive understanding of this definition, and also allows you to get acquainted with all the most important consequences of this function.
The first part of the presentation briefly gives the concept of the logarithm, and also demonstrates the construction of a graph based on it. After that comes the definition that needs to be learned, which is confirmed by the exclamation mark icon in the corner of the red box.
After restoring knowledge on a previously studied topic, students are invited to familiarize themselves with three identical equations that can be easily proved by any student who has to operate with such concepts 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 get the opportunity to apply their new knowledge in practice, thanks to the consideration of the solution of one example.
The presentation ends with another 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 an ordinary school algebra course - it is enough for the student to memorize, understand and be able to apply it when solving thematic examples.
Unlike the usual algebra course that a school textbook offers, 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 loading his brain and giving him the opportunity to take a break from a change in mental activity.
A quick understanding of the solutions of the proposed examples is facilitated by an interesting concept of presenting information, which is very difficult to find in a regular 11th grade algebra textbook. In the tasks proposed for consideration in the presentation, the most important data is highlighted in red or circled. This technique allows not only to quickly assimilate the most important information, but also teaches the student to independently search the right material from the whole context.
The section of modern algebra "properties of logarithms" is one of the most important in the entire course, as it provides the foundation for further, in-depth study of mathematics, which is necessary for hundreds of modern professions relating to various areas of human life. It is for this reason that you should not pass by 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 “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 finished look on a single page, so the lesson can not only be shown using various modern devices, but also simply printed out if the school has no other options.
JOHN NEPER (1550-1617)
Scottish mathematician -
inventor of logarithms.
In the 1590s came up with the idea
logarithmic calculations
and made the first tables
logarithms, but its famous
the work “Description of the 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 in connection with the needs of computational practice.
- In those days, to solve the problems of astronomy and navigation, very cumbersome calculations had to be made.
- The famous astronomer Johannes Kepler was the first to introduce the sign of the logarithm in 1624 - log. He used logarithms to find the orbit of Mars.
- The word "logarithm" is of Greek origin, which means - the ratio of numbers
0, and ≠1 is the exponent to which the number a must be raised to get b. "width="640" Definition
The logarithm of a positive number b to the base a, where a0, a ≠1 is 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 ; log2(1/2); log2(1/8);
log 3 27; log 3 81; log 3 3;
log 3 1; log3(1/9); log3(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 the logarithm
Calculate:
3 log 3 18 ; 3 5 log 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
Does not exist at
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) = log a b + log a c
( b
c )
a log a (b.c.) =
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; b0; 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,
a0; a ≠ 1; b0; c 0.
Example:
0; b0; 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. Logarithm of the degree with a positive base equal to degree multiplied by the logarithm of the base
log a b r = rlog a b
Example
a log a b =b
(a log a b ) r =b r
a rlog a b =b r
The formula for the transition from one base
logarithm to another, examples.
Definition of a derivative. Middle line. Investigation of a function for monotonicity. Works: Consolidation of the studied material. Calculate approximately using the differential. The smallest values of functions. Derivative and its application in algebra, geometry. The function in question. Task. Inequality. Signs of increasing and decreasing function. Dot. Definition. Finding the differential. Proof of inequalities.
""Integral" Grade 11" - How defeated you lay with the usual number on the page. Integral in Literature. A definite integral, you began to dream of me at night. Compose a phrase. What happiness I knew in the choice of the primitive. Zamyatin Evgeny Ivanovich (1884-1937). Find antiderivatives for functions. Epigraph. The novel "We" (1920). A series of substitutions and substitutions led to the solution of the problem. Illustration for the novel "We". Integral. Integral Group. Algebra lesson and started analysis.
"The use 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 magnitude of Capella (m1 = +0.2m) and Deneb (m2 = +1.3m). The unit of loudness. Stars, noise and logarithms. The harmful effects of industrial noise on the health of workers and labor production. Topic: "LOGARIFMS IN ASTRONOMY". Neper (1550 - 1617) and the Swiss I. Burgi (1552 - 1632).
""Functions" algebra" - Calculate. Let's make a table. Investigation of functions and construction of their graphs. The concept of the integral. The function F is called the antiderivative for the function f. Area of a curvilinear trapezoid. A function is an antiderivative for a function. Calculate the area S of the curvilinear trapezoid. "Integral from a to b ef from x de x". interval method. Find the intersection points of the graph with Ox (y = 0). Differentiation rules. Find the largest and smallest value functions on the segment.
"Examples of logarithmic inequalities" - Getting ready for the exam! Which functions are increasing and which are decreasing? Summary of the lesson. Find the right solution. Increasing. Algebra 11th grade. Task: solve the logarithmic inequalities proposed in the tasks of the USE-2010. Good luck on the USE! Cluster to fill during the lesson: Lesson objectives: Find the domain of the function. Between the numbers m and n put the sign > or<.(m, n >0). Graphs of logarithmic functions.
"The geometric meaning of the derivative of a function" - The value of the derivative of a function. Algorithm for compiling the tangent equation. geometric sense derivative. Equation of a straight line with a slope. Tangent Equations. Make a couple. Secant. Lesson vocabulary. I got it all. Correct mathematical idea. Calculation results. The limit position of the secant. Definition. Find the slope. Write the equation for the tangent to the graph of the function.
Lesson Objectives:
- Development of skills to systematize, generalize the properties of logarithms; apply them when simplifying expressions.
- Development of Conscious Perception educational material, visual memory, mathematical speech of students, to form the skills of self-learning, self-organization and self-esteem, to promote the development creative activity students.
- Education of cognitive activity, to instill in students love and respect for the subject, to teach them to see in it not only rigor, complexity, but also logic, simplicity and beauty.
Equipment:
- Interactive Whiteboard (StarBoard Software)
- Computers
- Presentation 1"Logarithms. Properties of logarithms»
- Presentation 2"Logarithms and Music"
- Technological map of the lesson
Lesson type: a lesson on the 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 “It is 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 see the Portrait of Pythagoras, notes and logarithms. What unites them? (Slide 2 on an interactive whiteboard or slide 2-3 in a presentation 1).
3. Logarithms in music
(Slide 3 on an interactive whiteboard or slide 4 in a presentation 1).
In his poem "Physicists and Lyrics" the poet Boris Slutsky wrote.
Even the fine arts feed on it.
Isn't the musical scale a set of advanced logarithms?
(Student's message - presentation attached)
4. The topic of the lesson(Slide 4 on the interactive whiteboard or slide 5 in the presentation 1). The class is divided into three groups, each student has a technological map.
II. Repetition
| 1 group | 2 group | 3 group |
| 1. Repetition of the theory | ||
Insert missing words: The logarithm of a numberb By………………………. but it is called …………….. the degree to which you need……………. base a to get a numberb . raise, base, indicator |
In the technological map of the lesson - Task 1 Collect the definition of a logarithm on a computer |
In the technological map of the lesson - Task 1 Write down the definition of the logarithm in mathematical language. |
| 2. Self-examination (Slide 5 on the interactive whiteboard or slide 7 in 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 the arrows on the computer to connect the formulas |
Task 2. In the technological map of the lesson, use the arrows to connect the formulas  |
Task 2. In the technological map of the lesson, complete the formulas  |
| 4. Peer review (Slide 8 on the interactive whiteboard or slide 10 in presentation 1) | ||
| 5. Applying properties | ||
| a) Orally (Slide 9-10 on the interactive whiteboard or slide 11-12 of presentation 1) Calculate and match answers 
|
||
| b) Find the mistakes (Slide 11 on the interactive whiteboard or slide 13 in Presentation 1) 
|
||
| c) Work in groups | ||
| Blackboard work. Calculate  |
Running a test in a routing Calculate:  |
Running 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 in Presentation 1) | ||
Calculate:  |
||
| 8. Sophism (Slide 14 on the interactive whiteboard or slide 16 in 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 sophism 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 in the presentation 1) | ||
Let's start with the inequality , which is indisputably true. Then comes the transformation  also beyond doubt. Greater value corresponds to a larger logarithm, so  , 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 in the presentation 1)
“Music can elevate or soothe the soul,
Painting is pleasing to the eye,
Poetry - to awaken feelings,
Philosophy - to satisfy the needs of the mind,
Engineering - improve the material side people's lives,
A mathematics can achieve all these goals.”
So said the American mathematician Maurice Kline.
Thank you for your work!
slide 2
Lesson Objectives:
Educational: Review the definition of the 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 the base a, where a > 0 and a ≠ 1, is the exponent to which you need to raise the number a to get the number b. Basic logarithmic identity alogab=b (where a>0, a≠1, b>0)
slide 4
The history of the emergence of logarithms
The word logarithm comes from two Greek words and it is translated as a ratio of numbers. During the sixteenth century the amount of work associated with carrying out approximate calculations in the course of solving various problems has sharply increased, and first of all, problems of astronomy, which has a direct practical use(when determining the position of ships by the stars and by the Sun). The biggest 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 unusually quickly entered into practice. The inventors of logarithms did not limit themselves to the development of a new theory. A practical tool was created - tables of logarithms - which dramatically increased the productivity of calculators. We 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 logarithmic ruler, which has become a working tool for many generations. The first tables of logarithms were compiled independently by the Scottish mathematician J. Napier (1550 - 1617) and the Swiss I. Burgi (1552 - 1632). Napier's tables included the values of the logarithms of sines, cosines and tangents for angles from 0 to 900 in increments 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, having reduced the work of the astronomer, extended his life. PS 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
degree properties
ax ay = ax + y = ax –y (x)y = ax y
Slide 8
Calculate:
Slide 9
Check:
Slide 10
PROPERTIES OF LOGARITHMS
slide 11
Application of the studied material
a) log 153 + log 155 = log 15(3 5) = log 1515 = 1, b) log 1545 - log 153 = log 15 = log 1515 = 1 c) log 243 = log 226 = 6 log 22 = 6, d) log 7494 = log 7(72)4 = log 7 78 = 8 log 77 = 8. 93; #290,291 - 294, 296* (odd examples)
slide 12
Find the second half of the formula
slide 13
Check:
Slide 14
Homework: 1. Learn the properties of logarithms 2. Textbook: § 16 pp. 92-93; 3. Task book: No. 290,291,296 (even examples)
slide 15
Continue the phrase: “Today in the lesson I learned ...” “Today in the lesson I learned ...” “Today in the lesson I met ...” “Today in the lesson I repeated ...” “Today in the lesson I fixed ...” The lesson is over!
slide 16
Used textbooks and teaching aids: Mordkovich A.G. Algebra and the beginnings of analysis. Grade 11: profile level textbook / A.G. Mordkovich, P.V. Semenov and others - M.: Mnemozina, 2007. Mordkovich A.G. Algebra and the beginnings of analysis. Grade 11: problem book of the profile level / A.G. Mordkovich, P.V. Semenov and others - M.: Mnemozina, 2007. Methodological literature used: Mordkovich A.G. Algebra. 10-11: Toolkit for the teacher. - M.: Mnemosyne, 2000 (Kaliningrad: Amber Tale, GIPP). Mathematics. Weekly supplement to the newspaper "The First of September".
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