Topic: Properties of logarithms.
Goals: 1. Educational: the formation of the ability to perform identical transformations,
using the properties of logarithms.
2. Developing goals: development of independent thinking, skills
justify your decision.
3. Educational goals: to promote the education of the cognitive need
students by creating a problem situation.
Basic concepts: logarithm of the product,
logarithm of the quotient, logarithm of the degree.
Independent student activity: solving problems on the topic "Properties of logarithms"
Fundamental question: Is it possible without them?
Problematic question:
Updating.(3 minutes.)
French writer Anatole France (1844-1924) remarked: “Learning can only be fun. To digest knowledge, one must absorb it with appetite. "
Let's follow the advice of the writer: we will be active in the lesson, attentive, we will "absorb" knowledge with great desire.
The task is this: to learn how to solve logarithmic expressions using the properties of logarithms.
1. Discussion No. 180 (3) from home. Tasks
log 0.2 log 2 (2x + 3)
log 0.2 log 2 (2x + 3) log 0.2 5
log 2 (2x + 3) log 2 32
Calculate:
a) log 1/3 1/3 c) log 1/3 1/9 e) log 1/3 9
b) log 1/3 3 d) log 1/3 1 f) log 1/3
3.Specify the scope of the function:
a) y = log 3 x c) y = log 3 | x |
b) y = log 3 (x-1) d) y = log 3 (-x)
4. Determine the nature of the monotonicity of the function:
a) y = log 3 x b) y = log 1/3 x c) y = -log 5 x
Learning new material.(10 minutes.)
Problematic question:
How to deduce properties of logarithms using power properties?
a x = b x = log a b
a y = c y = log a c
bc = a x b y = a log a b a log a c = a log a b + log a c
log a (bc) = log a b + log a c
Similarly, you can get the logarithm of the quotient and the degree:
log a b / c = log a b- log a c
log a b p = p log a b
Transition to the logarithm with a new base.
log a b = x, a x = b (logarithm)
log c a x = log c b
x log c a = log c b
x = log c b / log c a
log a p b = 1 / p log a b (removal of the exponent of the degree of base)
(Enter the formulas in the table)
| Properties of logarithms | Property name and wording |
| The logarithm of the product is equal to the sum of the logarithms |
|
| The logarithm of the quotient is equal to the difference of the logarithms |
|
| log a b p = p log a b | The logarithm of the power is equal to the product of the exponent degree by the logarithm of the base of this degree |
Students copy the table in their notebooks.
| Logarithms with the same grounds | Logarithms with different grounds |
| log a (bc) = log a b + log a c log a b / c = log a b - log a c log a b p = p log a b | log a b = log c b / log c a log a p b = 1 / p log a b |
Iii. Application. (20 minutes.)
No. 182 (1-5) (students analyze assignments for the possibility of using
properties of logarithms)
log 6 2+ log 6 3
log 1/15 25 + log 1/15 9
log 3 12 - log 3 4
log 2 12+ log 0.5 3
log 3 18 + log 1/3 2
Questions to this number:
Are the bases of the logarithms in the problem the same?
What part of the table will you work with?
Which formula do you use from the table?
What do you get as a result?
Write down the calculations.
the corresponding formula, name the resulting expressions and its
meaning.
No. 183 (1,2) - frontally.
Knowing that log 6 2 = a express through the expression 1) log 6 16
No. 183 (3.4) - independently.
(Answers: in 3) 7.5a; c 4) -4a)
No. 183 (5) - frontally
log 2 6 = log 6 6 / log 6 2 = 1 / a
(Students should note that this logarithm has a different base and, using the result of this task, get another formula log a b = 1 / log b a)
Textbook work: example # 1.
log 2 x = 3-4log 2 + 3log 2 3
3- 4 log 2 + 3 log 2 3 = log 2 2 3 - log 2 () 4 + log 2 3 3 = log 2 2 3 3 3 / () 4 = log 2 8 * 3 3/3 2 =
Log 2 (8 * 3) = log 2 24
log 2 x = log 2 24, x = 24
From the considered example, students are introduced to the new term "potentiation" - finding a number using a known logarithm.
No. 185 (2) - independently
(Answer: a = 20.25)
IV... Homework: p. 11 (ex. 1); (1 minute.)
No. 181 (1) - derivation of the formula for the logarithm of the quotient
№ 182 (3,5,7 *)
V... Lesson summary: (1 minute)
Conclusion: - what topic did you consider?
What was the task in the lesson?
What properties of logarithms do you know?
What is the logarithm of the product?
What is the logarithm of the quotient?
What is the logarithm of the power?
Grading with explanation.
VI... Informational resources:
G. K. Muravin, O. V. Muravina
Algebra and the beginning of analysis.
G. K. Muravin, O. V. Muravina
Algebra and the beginning of analysis. Textbook 10kl. M .: Bustard, 2004.
A. Ya.Simonov and others.
The system of training tasks and exercises in mathematics. M .: Education, 1998.
v... Crossnumber. (translated from English - cross numbers) - one of the types
number puzzles.
Lesson topic: Logarithms and their properties.
The purpose of the lesson:
- Educational- to form the concept of a logarithm, study the basic properties of logarithms and contribute to the formation of the ability to apply the properties of logarithms when solving problems.
- Developing - develop logical thinking; calculation technique; ability to work rationally.
- Educational - to contribute to the fostering of interest in mathematics, to foster a sense of self-control, responsibility.
Lesson type : Lesson in learning and primary consolidation of new knowledge.
Equipment: computer, multimedia projector, presentation "Logarithms and their properties", handouts.
Textbook: Algebra and the beginnings of mathematical analysis, 10-11. Sh.A. Alimov, Yu.M. Kolyagin et al., Education, 2014.
During the classes:
1. Organizational moment:checking the readiness of students for the lesson.
2. Repetition of the passed material.
Teacher questions:
1) Give a definition of the degree. What is called baseline and metric? (Nth root of the number a is a number whose n-th power is equal to a . 3 4 = 81.)
2) Formulate the properties of the degree.
3. Studying a new topic.
The topic of today's lesson is Logarithms and their properties (open your notebooks and write down the date and subject).
In this lesson we will get acquainted with the concept of "logarithm", we will also consider the properties of logarithms.
Let's ask a question:
1) To what degree do you need to raise 5 to get 25? Obviously, the second. The exponent to which the number 5 must be raised to get 25 is 2.
2) To what degree does 3 need to be raised to get 27? Obviously in the third. The exponent to which you need to raise the number 3 to get 27 is 3.
In all cases, we looked for an indicator of the degree to which something needs to be raised in order to get something. The exponent to which something needs to be raised is called the logarithm and is denoted log.
The number that we raise to the power, i.e. the base of the degree is called the base of the logarithm and is written in subscript. Then the number that we receive is written, i.e. the number we're looking for: log 5 25 = 2
This entry reads like this: "Logarithm base 5 of 25". Logarithm base 5 of 25 is the exponent to which 5 must be raised to get 25. This exponent is 2.
Let's look at the second example in a similar way.
Let us give the definition of a logarithm.
Definition . Logarithm of the number b> 0 with base a> 0, a ≠ 1 is called the exponent to which the number must be raised a, to get the number b.
Logarithm of the number b base a is denoted log a b.
The history of the emergence of the logarithm:
Logarithms were introduced by the Scottish mathematician John Napier (1550-1617) and the mathematician Jost Burghi (1552-1632).
Burghi came to logarithms earlier, but published his tables with a delay (in 1620), and the first in 1614. Napier's work "Description of the amazing table of logarithms" appeared.
From the point of view of computational practice, the invention of logarithms can be safely put alongside other, more ancient, great invention - our decimal numbering system.
A dozen years after the appearance of Napier's logarithms, the English scientist Gunther invented a very popular calculating device - the slide rule. She helped astronomers and engineers with calculations, she made it possible to quickly receive an answer with sufficient accuracy in three significant figures. Now it has been supplanted by calculators, but without the slide rule, neither the first computers nor microcalculators would have been created.
Let's consider some examples:
log 3 27 = 3; log 5 25 = 2; log 25 5 = 1/2;
Log 5 1/125 = -3; log -2 (-8) - does not exist; log 5 1 = 0; log 4 4 = 1
Consider the following examples:
ten . log a 1 = 0, a> 0, a ≠ 1;
twenty . log a а = 1, а> 0, a ≠ 1.
These two formulas are properties of the logarithm. They can be used to solve problems.
How to go from logarithmic to exponential equality? log a b = c, c - this is the logarithm, the exponent to which you want to raise a to get b. Therefore, a of degree c is equal to b: a c = b.
We derive the basic logarithmic identity: a log a b = b. (The teacher gives the proof on the chalkboard.)
Let's look at an example.
5 log 5 13 = 13
Let's consider some more important properties of logarithms.
Logarithm properties:
3 °. log a xy = log a x + log a y.
4 °. log a x / y = log a x - log a y.
5 °. log a x p = p log a x, for any real p.
Consider an example for checking 3 properties:
log 2 8 + log 2 16 = log 2 8 ∙ 16 = log 2 128 = 7
3 +4 = 7
Consider an example for checking property 5:
3 ∙ log 2 8 = log 2 8 3 = log 2 512 = 9
3∙3 = 9
4. Fastening.
Exercise 1. Name the property that is applied when calculating the following logarithms, and calculate (orally):
- log 6 6
- log 0.5 1
- log 6 3+ log 6 2
- log 3 6- log 3 2
- log 4 4 8
Task 2.
Here are 8 solved examples, some of which are correct, the rest with an error. Determine the correct equality (state its number), correct the errors in the rest.
- log 2 32+ log 2 2 = log 2 64 = 6
- log 5 5 3 = 2;
- log 3 45 - log 3 5 = log 3 40
- 3 ∙ log 2 4 = log 2 (4 ∙ 3)
- log 3 15 + log 3 3 = log 3 45;
- 2 ∙ log 5 6 = log 5 12
- 3 ∙ log 2 3 = log 2 27
- log 2 16 2 = 8.
Lesson on the topic "Logarithm, its properties".
Chertikhina L.P.
teacher
GB POU "VPT"
"Take as much as you can and want,
but not less obligatory. "
Lesson objectives:
know and be able to write down the definition of the logarithm, the basic logarithmic identity;
be able to apply the definition of the logarithm and the basic logarithmic identity when solving exercises;
get acquainted with the properties of logarithms;
learn to distinguish the properties of logarithms by their recording;
learn to apply the properties of logarithms when solving problems;
reinforce computational skills;
continue working on mathematical speech.
to form skills of independent work, work with a textbook, skills of independent acquisition of knowledge;
develop the ability to highlight the main thing when working with text;
to form the independence of thinking, mental operations: comparison, analysis, synthesis, generalization, analogy;
show students the role of systematic work to deepen and improve the strength of knowledge, on the culture of completing assignments;
develop the creativity of students.
Lesson type: communication of new knowledge.
Time spending: 1,5 hour
Equipment:
logarithm property table
task cards;
Teacher's PC, multimedia projector;
Lesson planOrganizing time. 1 minute.
Goal setting. 1 minute.
Verification of previously studied material 5 min
Introduction of the concept of a logarithm.
Definition of the logarithm. 5 minutes
6.Historical reference 10 min
Basic logarithmic identity. 10 min
Basic properties of logarithms 10 min
Generalization and systematization of knowledge. 7 minutes
Homework. 1 minute.
Creative application of knowledge, skills and abilities. 25 minutes
Summarizing. 5 minutes.
Guys, today in the lesson you have to test your ability to solve the simplest exponential equations so that you can introduce a concept that is new for you, then we will get acquainted with the properties of the new concept; you must learn to distinguish between these properties by their writing; learn to apply these properties when solving problems.
Be collected, alert and observant. Good luck!
Verification of previously studied material.Students are encouraged to determine the topic of the lesson by solving the 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
- Name the new concept with which we will get acquainted:
- The topic of our lesson is “Logarithm and its properties”. Try to find the root of the equation 2 x = 5. We can write the answer to this equation using a new concept. Read the slide text and write down the root of the equation.
4.1. Definition of the logarithm(slides 5-7)The logarithm of a positive number b to the base a, where a0, a ≠ 1 is the exponent to which a must be raised to get the number b.
1) log 10 100 = 2, because 10 2 = 100 (definition of the logarithm and properties of the degree),
2) log 5 5 3 = 3, because 5 3 = 5 3 (...),
3) log 4 = –1, because 4 -1 = (...).
In recording b = at number a is the basis of the degree, t- an indicator, b- degree. Number t -it is the exponent to which the base of a must be raised to get the number b. Hence, t is the logarithm of the number b by reason a: t = log
a
b
.
Substituting in equality t = logab expression b in the form of a degree, we get one more identity:
log a a t = t .
We can say that the formulas at= b and t = logab are equivalent, express the same relationship between numbers a, b and t(at a0, a 1, b0). Number t- arbitrarily, no restrictions are imposed on the exponent.
Substituting into equality at= b record number t in the form of a logarithm, we obtain an equality called basic logarithmic identity
:
= b
.
1) (3 2) log 3 7 = (3 log 3 7) 2 = 7 2 = 49 (degree of degree, basic logarithmic identity, definition of degree),
2) 7 2 log 7 3 = (7 log 7 3) 2 = 3 2 = 9 (...),
3) 10 3 log 10 5 = (10 log 10 5) 3 = 5 3 = 125 (...),
4) 0.1 2 log 0.1 10 = (0.1 log 0.1 10) 2 = 10 2 = 100 (...).
You've done a great job with the examples. Now calculate the following tasks written on the board:
a) log 15 3 + log 15 5 = ...,
b) log 15 45 - log 15 3 = ...,
c) log 4 8 = ...,
d) 7 =….
What do you think we need to know in order to perform actions with logarithms?
If students have difficulties, then ask the question: "To perform actions with degrees, what do you need to know?" (Answer: "Properties of the degree"). Re-ask the original question. (Properties of logarithms)
Here is a table with properties of logarithms. It is necessary to give a name to each property and formulate them correctly ”.
| Logarithm property name | Properties of logarithms |
|
| The logarithm of the unit. | log a 1 = 0, a 0, a 1. |
|
| Logarithm of the base. | log a a = 1, a 0, a 1. |
|
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 the 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 get the number b. The main logarithmic identity alogab = b (where a> 0, a ≠ 1, b> 0)
Slide 4
History of the origin of logarithms
The word logarithm comes from two Greek words and is translated as the 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 first of all, problems of astronomy, which has direct practical application (in determining the position of ships by the stars and by the Sun), has sharply increased. 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 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 - that 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 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. Burghi (1552 - 1632). Napier's tables include the values of the logarithms of sines, cosines and tangents for angles from 0 to 900 with a step of 1 minute. Burghi prepared his tables of logarithms of numbers, but they were published in 1620, after the publication of Napier's tables, and therefore remained unnoticed. Napier John (1550-1617)
Slide 6
The invention of logarithms, by reducing the work of the astronomer, prolonged his life. PS Laplace Therefore, the discovery of logarithms, which reduces multiplication and division of numbers to addition and subtraction of their logarithms, lengthened, according to Laplace's expression, 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
LOGARITHM PROPERTIES
Slide 11
Application of the studied material
a) log 153 + log 155 = log 15 (35) = 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. Page 93; No. 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. Problem 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 reinforced ...” The lesson is over!
Slide 16
Used textbooks and teaching aids: Mordkovich A.G. Algebra and the beginning of analysis. Grade 11: profile-level textbook / A.G. Mordkovich, P.V. Semenov et al. - M .: Mnemozina, 2007. Mordkovich A.G. Algebra and the beginning of analysis. Grade 11: profile-level problem book / A.G. Mordkovich, P.V. Semenov et al. - M .: Mnemosina, 2007. Methodical literature used: Mordkovich A.G. Algebra. 10-11: teaching aid for the teacher. - M .: Mnemozina, 2000 (Kaliningrad: Amber Skaz, GIPP). Maths. Weekly supplement to the newspaper "First September".
Methodical development of a lesson in mathematics
"Logarithms and their properties"
The purpose of the lesson:
Educational- introduce the concept of a logarithm, study the basic properties of logarithms and contribute to the formation of the ability to apply the properties of logarithms when solving problems.
Developing- develop mathematical thinking; calculation technique; the ability to think logically and work rationally; promote the development of students' self-control skills.
Educational- to promote the fostering of interest in the topic, foster a sense of self-control, responsibility.
Lesson Objectives:
To develop students' skills to compare, contrast, analyze, draw independent conclusions.
Key competencies: the ability to independently search, extract, systematize, analyze and select information necessary for solving educational problems; the ability to independently master the knowledge and skills necessary to solve the task.
Lesson type: Lesson in learning and primary consolidation of new knowledge.
Equipment: computer, multimedia projector, presentation "Logarithms and their properties", handouts.
Keywords: logarithm; properties of the logarithm.
Software: MS Power Point.
Interdisciplinary connections: history.
Intra-subject communications: "Root of the n-th degree and their properties."
Lesson plan
Organizing time.
Repetition of the passed material.
Explanation of the new material.
Anchoring.
Independent work.
Homework. Summing up the lesson.
During the classes:
- Organizational moment: checking the readiness of students for the lesson; attendant's report .
Good afternoon, students.
I want to start this lesson with the words of A.N. Krylova: "Sooner or later any correct mathematical idea finds application in this or that case."
Repetition of the passed material.
Students are encouraged to remember:
What is degree, base and exponent.
Nth root of a number a is a number whose n-th power is equal to a. 3 4 = 81.
2) Basic properties of degrees.
3. Post a new topic.
Now let's move on to a new topic. The topic of today's lesson is the Logarithm and their properties (open your notebooks and write down the date and subject).
In this lesson we will get acquainted with the concept of "logarithm", we will also consider the properties of logarithms. This topic is relevant, because the logarithm is always found in the final certification in mathematics.
Let's ask a question:
1) To what degree does 3 need to be raised to get 9? Obviously, the second. The exponent to which you need to raise the number 3 to get 9 is 2.
2) To what degree does 2 need to be raised to get 8? Obviously, the second. The exponent to which you need to raise the number 2 to get 8 is 3.
In all cases, we looked for an indicator of the degree to which something needs to be raised in order to get something. The exponent to which something needs to be raised is called the logarithm and is denoted log.
The number that we raise to the power, i.e. the base of the degree is called the base of the logarithm and is written in subscript. Then the number that we receive is written, i.e. the number we're looking for: log 3 9=2
This entry reads like this: "Logarithm of 9 to base 3". Logarithm base 3 of 9 is the exponent to which 3 must be raised to get 9. This exponent is 2.
The second example is similar.
Let us give the definition of a logarithm.
Definition. Logarithm of the number b> 0 by reason a> 0, a ≠ 1 is called the exponent to which the number must be raiseda, to get the numberb .
Logarithm of the number b by reason a denoted log a b.
The history of the emergence of the logarithm:
Logarithms were introduced by the Scottish mathematician John Napier (1550-1617) and the mathematician Jost Burghi (1552-1632).
From the point of view of computational practice, the invention of logarithms, if possible, can be safely put alongside other, more ancient great inventions of the Indians - our decimal numbering system.
A dozen years after the appearance of Napier's logarithms, the English scientist Gunther invented a very popular calculating device - the slide rule.
She helped astronomers and engineers with calculations, she made it possible to quickly receive an answer with sufficient accuracy in three significant figures. Now it has been supplanted by calculators, but without the slide rule, neither the first computers nor microcalculators would have been built.
Let's consider some examples:
log 3 27 = 3; log 5 25 = 2; log 25 5 = 1/2; log 5 1/125 = -3; log -2 -8- does not exist; log 5 1 = 0; log 4 4 = 1
Consider the following examples:
ten . log a 1 = 0, a> 0, a ≠ 1;
twenty . log a а = 1, а> 0, a ≠ 1.
These two formulas are properties of the logarithm. Write down the properties and they need to be remembered.
In mathematics, the following abbreviation is accepted:
log 10 a =lga is the decimal logarithm of the number a(the letter "o" is skipped and base 10 is omitted).
log e a = lna - natural logarithm of the number a."E" is such an irrational number, equal to 2.7 (the letter "o" is omitted, and the base "e" is not put).
Let's consider some examples:
lg 10 = 1; lg 1 = 0
ln e = 1; ln 1 = 0.
How to go from logarithmic to exponential equality: log a b= s, s - this is the logarithm, the exponent to which you want to raise a, To obtain b... Hence, a degree with is equal to b: a with = b.
Consider five logarithmic equalities. Assignment: check their correctness. There are errors among these examples. For verification, we will use this scheme.
lg 1 = 2 (10 2 =100)- this equality is not true.
log 1/2 4 = 2- this equality is not true.
log 3 1=1 - this equality is not true.
log 1/3 9 = -2 - this equality is true.
log 4 16 = -2- this equality is not true.
We derive the basic logarithmic identity: a log a b = b
Let's look at an example.
5 log 5 13 =13
Logarithm properties:
3 °. log a xy = log a x + log a y.
4 °. log a x / y = log a x - log a y.
5 °. log ax p = p log ax, for any real p.
Consider an example for checking 3 properties:
log 2 8 + log 2 32 = log 2 8 ∙ 32 = log 2 256 = 8
Consider an example for checking property 5:
3∙ log 2 8= log 2 8 3 = log 2 512 =9
3∙3 = 9
The formula for the transition from one base of a logarithm to another base:
You will need this formula when calculating the logarithm using the calculator. Let's take an example: log 3 7 = lg7 / lg3. The calculator can only calculate the decimal and natural logarithm. Enter the number 7 and press the "log" button, also enter the number 3 and press the "log" button, divide the upper value by the lower one and get the answer.
- Anchoring.
- log 6 6
Here are 8 solved examples, some of which are correct, the rest with an error. Determine the correct equality (state its number), correct the errors in the rest.
- log 2 32+ log 2 2 = log 2 64 = 6
log 5 5 3 = 2;
log 3 45 - log 3 5 = log 3 40
3 ∙ log 2 4 = log 2 (4 ∙ 3)
log 3 15 + log 3 3 = log 3 45;
2 ∙ log 5 6 = log 5 12
3 ∙ log 2 3 = log 2 27
log 2 16 2 = 8.
- ZUN check - independent work on the cards.
- log 4 16 log 25 125 log 8 2 log 6 6
- log 3 27 log 4 8 log 49 7 log 5 5
Summarizing. Homework. Grading.
