Converting expressions using the properties of logarithms, examples, solutions. Identical transformations of exponential and logarithmic expressions B4 identical transformations of logarithmic expressions

OPEN ALGEBRA LESSON IN 11th CLASS

LESSON TOPIC

"CONVERTING EXPRESSIONS,

CONTAINING LOGARITHMES"

Lesson objectives:

repeat the definition of the logarithm of a number, the basic logarithmic identity;

consolidate the basic properties of logarithms;

strengthen the practical orientation of this topic for high-quality preparation for the UNT;

promote solid assimilation of the material;

promote the development of self-control skills in students.

Lesson type: combined using an interactive test.

Equipment: projector, screen, posters with tasks, answer sheet.

Lesson plan:

Organizing time.

Updating knowledge.

Interactive test.

"Tournament with logarithms"

Solving problems according to the textbook.

Summarizing. Filling out the answer sheet.

Grading.

During the classes

1. Organizational moment.

2. Determining the objectives of the lesson.

Hello guys! Today we have an unusual lesson, a lesson - a game, which we will conduct in the form of a tournament with logarithms.

Let's start the lesson with an interactive test.

3. Interactive test:

4. Tournament with logarithms:

Definition of logarithm.

Logarithmic identities:

Simplify:

Find the meaning of the expression:

Properties of logarithms .

Conversion:

Working with the textbook.

Summarizing.

Students fill out their own answer sheet.

Give marks for each answer.

Grading. Homework. Annex 1.

Today you are immersed in logarithms,

They must be calculated accurately.

Of course, you will meet them in the exam,

We can only wish you success!

I option

a) 9 ½ =3; b) 7 0 =1.

A)log8=6; b)log9=-2.

a) 1.7 log 1,7 2 ; b) 2 log 2 5 .

4. Calculate:

A) lg8+lg125;

b)log 2 7-log 2 7/16

V)log 3 16/log 3 4.

II option

1. Find the logarithm to base a of a number represented as a power with base a:

a) 32 1/5 =2; b) 3 -1 =1/3.

2. Check the validity of the equality:

A)log27=-6; b)log 0,5 4=-2.

3. Simplify the expression using the basic logarithmic identities:

a) 5 1+ log 5 3 ; b) 10 1- lg 2

4. Calculate:

A)log 12 4+log 12 36;

b) lg13-lg130;

V) (lg8+lg18)/(2lg2+lg3).

III option

1. Find the logarithm to base a of a number represented as a power with base a:

a) 27 2/3 =9; b) 32 3/5 =8.

2. Check the validity of the equality:

A)log 2 128=;

b)log 0,2 0,008=3.

3. Simplify the expression using the basic logarithmic identities:

a) 4 2 log 4 3 ;

b) 5 -3 log 5 1/2 .

4. Calculate:

A)log 6 12+log 6 18;

b)log 7 14-log 7 6+log 7 21;

V) (log 7 3/ log 7 13)∙ log 3 169.

IV option

1. Find the logarithm to base a of a number represented as a power with base a:

a) 81 3/4 =27; b) 125 2/3 =25.

2. Check the validity of the equality:

A)log √5 0,2=-2;

b)log 0,2 125=-3.

3. Simplify the expression using the basic logarithmic identities:

a) (1/2) 4 log 1/2 3 ;

b) 6 -2 log 6 5 .

4. Calculate:

A)log 14 42-log 14 3;

b)log 2 20-log 2 25+log 2 80;

V)log 7 48/ log 7 4- 0,5 log 2 3.

Transnistrian State University

them. T.G. Shevchenko

Faculty of Physics and Mathematics

Department of Mathematical Analysis

and methods of teaching mathematics

COURSE WORK

"Identity transformations

exponential and logarithmic

expressions"

Work completed:

student of _______ group

Faculty of Physics and Mathematics

_________________________

I checked the work:

_________________________

Tiraspol, 2003

Introduction………………………………………………………………………………2

Chapter 1. Identical transformations and teaching methods in the school course of algebra and beginning of analysis……………………………………..4

§1. Formation of skills in applying specific types of transformations……………………………………………………………………………………….4

§2. Features of the organization of a knowledge system in the study of identity transformations.…….………………………….………..………….5

§3. Mathematics program……………………………………………………….11

Chapter 2. Identical transformations and calculations of exponential and logarithmic expressions……………………………...…………………13

§1. Generalization of the concept of degree……………………………………..13

§2. Exponential function……………………………………………………..15

§3. Logarithmic function…………………………………….16

Chapter 3. Identical transformations of exponential and logarithmic expressions in practice..........................................................................19

Conclusion……………………………………………………………..24

List of references……………………………………………………….25
Introduction

In this course work, the identical transformations of exponential and logarithmic functions will be considered, and the methodology for teaching them in a school algebra course and the beginning of analysis will be considered.

The first chapter of this work describes the methodology for teaching identity transformations in a school mathematics course, and also includes a mathematics program in the course “Algebra and beginnings of analysis” with the study of exponential and logarithmic functions.

The second chapter directly examines the exponential and logarithmic functions themselves, their basic properties used in identity transformations.

The third chapter is solving examples and problems using identical transformations of exponential and logarithmic functions.

Studying various transformations of expressions and formulas takes up a significant part of the teaching time in a school mathematics course. The simplest transformations, based on the properties of arithmetic operations, are carried out already in elementary school and in grades IV-V. But the main burden of developing the skills and abilities to carry out transformations is borne by the school algebra course. This is due both to the sharp increase in the number and variety of transformations being carried out, and to the complication of activities to substantiate them and clarify the conditions of applicability, to the identification and study of the generalized concepts of identity, identical transformation, equivalent transformation, logical consequence.

The culture of performing identity transformations develops in the same way as the culture of calculations, based on solid knowledge of the properties of operations on objects (numbers, vectors, polynomials, etc.) and algorithms for their implementation. It manifests itself not only in the ability to correctly substantiate transformations, but also in the ability to find the shortest path to transition from the original analytical expression to the expression that most corresponds to the purpose of the transformation, in the ability to monitor changes in the domain of definition of analytical expressions in a chain of identical transformations, in the speed and accuracy of performing transformations .

Ensuring a high culture of calculations and identity transformations is an important problem in teaching mathematics. However, this problem is still far from being satisfactorily solved. Proof of this is the statistical data of public education authorities, which annually record errors and irrational methods of calculations and transformations made by students of various classes when performing tests. This is confirmed by feedback from higher educational institutions about the quality of mathematical knowledge and skills of applicants. One cannot but agree with the conclusions of public education authorities and universities that the insufficiently high level of culture of calculations and identical transformations in secondary school is a consequence of formalism in students’ knowledge, the separation of theory from practice.

Chapter 1.

Identical transformations and teaching methods

in the school course of algebra and beginning of analysis.

§1. Formation of application skills

specific types of transformationtitles.

The system of techniques and rules for carrying out transformations used at the stage of beginning algebra has a very wide range of applications: it is used in the study of the entire mathematics course. However, precisely because of its low specificity, this system requires additional transformations that take into account the structural features of the expressions being transformed and the properties of newly introduced operations and functions. Mastering the corresponding types of transformations begins with the introduction of abbreviated multiplication formulas. Then transformations associated with the operation of exponentiation are considered with various classes of elementary functions - exponential, power, logarithmic, trigonometric. Each of these types of transformations goes through a learning phase in which attention is focused on mastering their characteristic features.

As material accumulates, it becomes possible to highlight the common features of all the transformations under consideration and, on this basis, introduce the concepts of identical and equivalent transformations.

It should be noted that the concept of identity transformation is given in the school algebra course not in full generality, but only in application to expressions. Transformations are divided into two classes: identical transformations are transformations of expressions, and equivalent transformations are transformations of formulas. In the case when there is a need to simplify one part of the formula, an expression is highlighted in this formula, which serves as an argument for the applied identity transformation. The corresponding predicate is considered unchanged.

Concerning organizing a holistic system of transformations(synthesis), then its main goal is to form a flexible and powerful; apparatus suitable for use in solving a variety of educational tasks.

In the course of algebra and the beginning of analysis, a holistic system of transformations, already formed in its main features, continues to gradually improve. Some new types of transformations are also added to it, but they only enrich it, expand its capabilities, but do not change its structure. The methodology for studying these new transformations is practically no different from that used in the algebra course.

§2. Features of the organizationtask systems

when studying identity transformations.

The basic principle of organizing any system of tasks is to present them from simple to complex, taking into account the need for students to overcome feasible difficulties and create problematic situations. This basic principle requires specification in relation to the features of this educational material. To describe various systems of tasks in mathematics methods, the concept is used cycle of exercises. The cycle of exercises is characterized by the combination in a sequence of exercises of several aspects of studying and techniques for arranging the material. In relation to identity transformations, the idea of ​​a cycle can be given as follows.

The cycle of exercises is associated with the study of one identity, around which other identities that are in a natural connection with it are grouped. The cycle, along with executive ones, includes tasks that require recognition of the applicability of the identity in question. The identity under study is used to carry out calculations on various numerical domains. The specificity of identity is taken into account; in particular, the figures of speech associated with it are organized.

The tasks in each cycle are divided into two groups. The first includes tasks performed during initial acquaintance with identity. They serve as educational material for several consecutive lessons united by one topic. The second group of exercises connects the identity being studied with various applications. This group does not form a compositional unity - the exercises here are scattered on various topics.

The described cycle structure refers to the stage of developing skills in applying specific types of transformations. At the final stage, the synthesis stage, the cycles are modified. Firstly, both groups of tasks are combined to form an “expanded” cycle, and the simplest ones in terms of wording or complexity of completing the task are excluded from the first group. The remaining types of tasks become more complex. Secondly, there is a merging of cycles related to different identities, due to which the role of actions to recognize the applicability of a particular identity increases.

Let us note the features of task cycles related to identities for elementary functions. These features are due to the fact that, firstly, the corresponding identities are studied in connection with the study of functional material and, secondly, they appear later than the identities of the first group and are studied using already formed skills for carrying out identity transformations.

Each newly introduced elementary function dramatically expands the range of numbers that can be designated and named individually. Therefore, the first group of cycle tasks should include tasks to establish connections between these new numerical domains and the original domain of rational numbers. Let's give examples of such tasks.

Example 1 . Calculate:

Next to each expression an identity is indicated, in the cycles for which the proposed tasks may be present. The purpose of such tasks is to master the features of records, including symbols of new operations and functions, and to develop mathematical speech skills.

A significant part of the use of identity transformations associated with elementary functions falls on the solution of irrational and transcendental equations. The cycles related to the assimilation of identities include only the simplest equations, but here it is advisable to carry out work on mastering the method of solving such equations: reducing it by replacing the unknown with an algebraic equation.

The sequence of steps for this solution is as follows:

a) find a function for which this equation can be represented in the form;

b) make the substitution and solve the equation;

c) solve each of the equations, where is the set of roots of the equation.

When using the described method, step b) is often performed implicitly, without introducing a notation for . In addition, students often prefer, from the various paths leading to finding an answer, to choose the one that leads to the algebraic equation faster and easier.

Example 2 . Solve the equation.

First way:

Second way:

Here you can see that with the first method step a) is more difficult than with the second. The first method is “more difficult to start with,” although the further course of the solution is much simpler. On the other hand, the second method has the advantages of greater ease and greater precision in learning to reduce to an algebraic equation.

For a school algebra course, typical tasks are in which the transition to an algebraic equation is even simpler than in this example. The main load of such tasks relates to the identification of step c) as an independent part of the solution process associated with the use of the properties of the elementary function being studied.

Example 3 . Solve the equation:

These equations are reduced to the equations: a) or ; b) or . To solve these equations, knowledge of only the simplest facts about the exponential function is required: its monotonicity, range of values. Like the previous example, equations a) and b) can be classified as the first group of a series of exercises for solving quadratic exponential equations.

Thus, we come to a classification of tasks in cycles related to solving transcendental equations that include an exponential function:

1) equations that reduce to equations of the form and have a simple, general answer: ;

2) equations that reduce to equations , where is an integer, or , where ;

3) equations that reduce to equations and require explicit analysis of the form in which the number is written .

Tasks for other elementary functions can be classified similarly.

A significant part of the identities studied in the courses of algebra and algebra and principles of analysis are proved in them or, at least, explained. This aspect of the study of identities is of great importance for both courses, since evidentiary reasoning in them is carried out with the greatest clarity and rigor precisely in relation to identities. Beyond this material, evidence is usually less complete; it is not always distinguished from the substantiation used.

The properties of arithmetic operations are used as the support on which proofs of identities are built.

The educational impact of calculations and identical transformations can be aimed at the development of logical thinking, if only students are systematically required to justify calculations and identical transformations, and at the development of functional thinking, which is achieved in various ways. The importance of calculations and identical transformations in the development of will, memory, intelligence, self-control, and creative initiative is quite obvious.

The demands of everyday and industrial computing practice require students to develop strong, automated skills in rational calculations and identity transformations. These skills are developed in the process of any computational work, however, special training exercises in fast calculations and transformations are necessary.

So, if the lesson involves solving logarithmic equations using the basic logarithmic identity, then it is useful to include in the lesson plan oral exercises on simplifying or calculating the meanings of expressions: , , . The purpose of the exercises is always communicated to students. During the exercise, it may be necessary to require students to justify individual transformations, actions, or the solution to an entire problem, even if this was not planned. Where different ways of solving a problem are possible, it is advisable to always ask questions: “How was the problem solved?”, “Who solved the problem in a different way?”

The concepts of identity and identity transformation are explicitly introduced in the VI grade algebra course. The very definition of identical expressions cannot be practically used to prove the identity of two expressions, and understand that the essence of identical transformations is to apply to the expression the definitions and properties of those actions that are indicated in the expression, or to add to it an expression that is identically equal to 0, or in multiplying it by an expression identically equal to one. But even having mastered these provisions, students often do not understand why these transformations allow us to assert that the original and resulting expressions are identical, i.e. take the same values ​​for any systems (sets) of variable values.

It is also important to ensure that students clearly understand that such conclusions of identical transformations are consequences of the definitions and properties of the corresponding actions.

The apparatus of identity transformations, accumulated in previous years, is expanded in grade VI. This extension begins with the introduction of an identity expressing the property of the product of powers with the same bases: , where , are integers.

§3. Mathematics program.

In the school course “Algebra and the beginnings of analysis,” students systematically study exponential and logarithmic functions and their properties, identical transformations of logarithmic and exponential expressions and their application to solving the corresponding equations and inequalities, and become familiar with basic concepts and statements.

In the 11th grade, algebra lessons take 3 hours a week, for a total of 102 hours a year. The program takes 36 hours to study exponential, logarithmic and power functions.

The program includes consideration and study of the following issues:

The concept of a degree with a rational exponent. Solving irrational equations. Exponential function, its properties and graph. Identical transformations of exponential expressions. Solving exponential equations and inequalities. Logarithm of a number. Basic properties of logarithms. Logarithmic function, its properties and graph. Solving logarithmic equations and inequalities. Derivative of an exponential function. Number and natural logarithm. Derivative of a power function.

The main purpose of the exponential and logarithmic function section is to familiarize students with exponential, logarithmic and power functions; teach students to solve exponential and logarithmic equations and inequalities.

The concepts of the th root and the degree with a rational exponent are a generalization of the concepts of the square root and the degree with an integer exponent. Students should pay attention to the fact that the properties of roots and powers with rational exponents considered here are similar to those properties possessed by the previously studied square roots and powers with integer exponents. It is necessary to devote enough time to practicing the properties of degrees and developing the skills of identity transformations. The concept of a degree with an irrational exponent is introduced on a visual and intuitive basis. This material plays an auxiliary role and is used when introducing the exponential function.

The study of the properties of exponential, logarithmic and power functions is constructed in accordance with the accepted general scheme for studying functions. In this case, an overview of the properties is given depending on the parameter values. Exponential and logarithmic inequalities are solved based on the studied properties of functions.

A characteristic feature of the course is the systematization and generalization of students’ knowledge, consolidation and development of skills acquired in the algebra course, which is carried out both when studying new material and when conducting generalized repetition.
Chapter 2.

Identity transformations and calculations

exponential and logarithmic expressions

§1. Generalization of the concept of degree.

Definition: The th root of a pure number is a number whose th power is equal to .

According to this definition, the th root of a number is the solution to the equation. The number of roots of this equation depends on and. Let's consider the function. As is known, on the interval this function increases for any value and takes all values ​​from the interval. According to the root theorem, the equation for any has a non-negative root and, moreover, only one. He is called arithmetic root of the th degree of a number and denote ; the number is called root index, and the number itself is radical expression. The sign is also called a radical.

Definition: Arithmetic root of the th power of a number is a non-negative number whose -th power is equal to .

For even numbers the function is even. It follows that if , then the equation, in addition to the root, also has a root. If , then there is one root: ; if , then this equation has no roots, since the even power of any number is non-negative.

For odd values, the function increases along the entire number line; its range is the set of all real numbers. Applying the root theorem, we find that the equation has one root for any and, in particular, for . This root for any value is denoted by .

For roots of odd degree, the equality holds. In fact, , i.e. number is the th root of . But such a root for odd is the only one. Hence, .

Note 1: For any real

Let us recall the known properties of arithmetic roots of the th degree.

For any natural number, integer and any non-negative integers and the equalities are valid:

Degree with a rational exponent.

The expression is defined for all and except the case at . Let us recall the properties of such powers.

For any numbers , and any integers and the equalities are valid:

We also note that if , then at and at .

Definition: A power of a number with a rational exponent, where is an integer and is a natural number, is called a number.

So, by definition.

With the formulated definition of a degree with a rational exponent, the basic properties of degrees are preserved, which are true for any exponents (the difference is that the properties are true only for positive bases).

§2. Exponential function.

Definition: The function given by the formula (where , ) is called exponential function with base .

Let us formulate the main properties of the exponential function.

Function graph (Fig. 1)

These formulas are called basic properties of degrees.

You can also notice that the function is continuous on the set of real numbers.

§3. Logarithmic function.

Definition: Logarithm numbers to the base is called the exponent to which the base must be raised. To get the number .

The formula (where , and ) is called basic logarithmic identity.

When working with logarithms, the following properties are used, resulting from the properties of the exponential function:

For any( )and any positive and equalities are satisfied:

5. for any real .

The basic properties of logarithms are widely used when converting expressions containing logarithms. For example, the formula for moving from one logarithm base to another is often used: .

Let be a positive number not equal to 1.

Definition: The function given by the formula is called logarithmic function with base.

Let us list the main properties of the logarithmic function.

1. The domain of definition of a logarithmic function is the set of all positive numbers, i.e. .

2. The range of values ​​of a logarithmic function is the set of all real numbers.

3. The logarithmic function throughout the entire domain of definition increases (at ) or decreases (at ).

Function graph (Fig. 2)

Graphs of exponential and logarithmic functions that have the same base are symmetrical with respect to a straight line(Fig. 3).

Chapter 3.

Identical transformations of exponential and

logarithmic expressions in practice.

Exercise 1.

Calculate:

Solution:

Answer:; ; ; ; .; , we get that

I considered methods for developing skills in students when studying this material. She also presented a program in mathematics for studying the course of exponential and logarithmic functions in the course “Algebra and the beginning of analysis.”

The work presented tasks of different complexity and content, using identical transformations. These tasks can be used to conduct tests or independent work to test students' knowledge.

The course work, in my opinion, was carried out within the framework of the methodology of teaching mathematics in secondary educational institutions and can be used as a visual aid for school teachers, as well as for full-time and part-time students.

List of used literature:

1. Algebra and the beginnings of analysis. Ed. Kolmogorova A.N. M.: Education, 1991.
2. Program for secondary schools, gymnasiums, lyceums. Mathematics 5-11 grades. M.: Bustard, 2002.
3. I.F. Sharygin, V.I. Golubev. Optional course in mathematics (problem solving). Uch. allowance for 11th grade. M.: Education, 1991.
4. V.A. Oganesyan et al. Methods of teaching mathematics in secondary school: General methods; A textbook for students of the Faculty of Physics and Mathematics of Pedagogical Institutes. -2nd edition revised and expanded. M.: Education, 1980.
5. Cherkasov R.S., Stolyar A.A. Methods of teaching mathematics in secondary school. M.: Education, 1985.
6. Magazine "Mathematics at school".

Transnistrian State University

them. T.G. Shevchenko

Faculty of Physics and Mathematics

Department of Mathematical Analysis

and methods of teaching mathematics

COURSE WORK

"Identity transformations

exponential and logarithmic

expressions"

Work completed:

student of _______ group

Faculty of Physics and Mathematics

_________________________

I checked the work:

_________________________

Tiraspol, 2003

Introduction………………………………………………………………………………2

Chapter 1. Identity transformations and teaching methods in the school course of algebra and beginning of analysis……………………………………..4

§1. Formation of skills in applying specific types of transformations……………………………………………………………………………………….4

§2. Features of the organization of a knowledge system in the study of identity transformations.…….………………………….………..………….5

§3. Mathematics program……………………………………………………….11

Chapter 2. Identical transformations and calculations of exponential and logarithmic expressions……………………………...………………13

§1. Generalization of the concept of degree……………………………………..13

§2. Exponential function……………………………………………………..15

§3. Logarithmic function…………………………………….16

Chapter 3. Identical transformations of exponential and logarithmic expressions in practice.................................................... ...................................19

Conclusion……………………………………………………………..24

List of references……………………………………………………….25
Introduction

In this course work, the identical transformations of exponential and logarithmic functions will be considered, and the methodology for teaching them in a school algebra course and the beginning of analysis will be considered.

The first chapter of this work describes the methodology for teaching identity transformations in a school mathematics course, and also includes a mathematics program in the course “Algebra and beginnings of analysis” with the study of exponential and logarithmic functions.

The second chapter directly examines the exponential and logarithmic functions themselves, their basic properties used in identity transformations.

The third chapter is solving examples and problems using identical transformations of exponential and logarithmic functions.

Studying various transformations of expressions and formulas takes up a significant part of the teaching time in a school mathematics course. The simplest transformations, based on the properties of arithmetic operations, are carried out already in elementary school and in grades IV–V. But the main burden of developing the skills and abilities to carry out transformations is borne by the school algebra course. This is due both to the sharp increase in the number and variety of transformations being carried out, and to the complication of activities to substantiate them and clarify the conditions of applicability, to the identification and study of the generalized concepts of identity, identical transformation, equivalent transformation, logical consequence.

The culture of performing identity transformations develops in the same way as the culture of calculations, based on solid knowledge of the properties of operations on objects (numbers, vectors, polynomials, etc.) and algorithms for their implementation. It manifests itself not only in the ability to correctly substantiate transformations, but also in the ability to find the shortest path to transition from the original analytical expression to the expression that most corresponds to the purpose of the transformation, in the ability to monitor changes in the domain of definition of analytical expressions in a chain of identical transformations, in the speed and accuracy of performing transformations .

Ensuring a high culture of calculations and identity transformations is an important problem in teaching mathematics. However, this problem is still far from being satisfactorily solved. Proof of this is the statistical data of public education authorities, which annually record errors and irrational methods of calculations and transformations made by students of various classes when performing tests. This is confirmed by feedback from higher educational institutions about the quality of mathematical knowledge and skills of applicants. One cannot but agree with the conclusions of public education authorities and universities that the insufficiently high level of culture of calculations and identical transformations in secondary school is a consequence of formalism in students’ knowledge, the separation of theory from practice.

Identical transformations and teaching methods

in the school course of algebra and beginning of analysis.

§1. Formation of application skills

specific types of transformations.

The system of techniques and rules for carrying out transformations used at the stage of beginning algebra has a very wide range of applications: it is used in the study of the entire mathematics course. However, precisely because of its low specificity, this system requires additional transformations that take into account the structural features of the expressions being transformed and the properties of newly introduced operations and functions. Mastering the corresponding types of transformations begins with the introduction of abbreviated multiplication formulas. Then transformations associated with the operation of exponentiation are considered with various classes of elementary functions - exponential, power, logarithmic, trigonometric. Each of these types of transformations goes through a learning phase in which attention is focused on mastering their characteristic features.

As material accumulates, it becomes possible to highlight the common features of all the transformations under consideration and, on this basis, introduce the concepts of identical and equivalent transformations.

It should be noted that the concept of identity transformation is given in the school algebra course not in full generality, but only in application to expressions. Transformations are divided into two classes: identical transformations are transformations of expressions, and equivalent transformations are transformations of formulas. In the case when there is a need to simplify one part of the formula, an expression is highlighted in this formula, which serves as an argument for the applied identity transformation. The corresponding predicate is considered unchanged.

As for the organization of an integral system of transformations (synthesis), its main goal is to form a flexible and powerful; apparatus suitable for use in solving a variety of educational tasks.

In the course of algebra and the beginning of analysis, a holistic system of transformations, already formed in its main features, continues to gradually improve. Some new types of transformations are also added to it, but they only enrich it, expand its capabilities, but do not change its structure. The methodology for studying these new transformations is practically no different from that used in the algebra course.

§2. Features of the organization of the task system

when studying identity transformations.

The basic principle of organizing any system of tasks is to present them from simple to complex, taking into account the need for students to overcome feasible difficulties and create problematic situations. This basic principle requires specification in relation to the features of this educational material. To describe various systems of tasks in mathematics methods, the concept of a cycle of exercises is used. The cycle of exercises is characterized by the combination in a sequence of exercises of several aspects of studying and techniques for arranging the material. In relation to identity transformations, the idea of ​​a cycle can be given as follows.

The cycle of exercises is associated with the study of one identity, around which other identities that are in a natural connection with it are grouped. The cycle, along with executive ones, includes tasks that require recognition of the applicability of the identity in question. The identity under study is used to carry out calculations on various numerical domains. The specificity of identity is taken into account; in particular, the figures of speech associated with it are organized.

The tasks in each cycle are divided into two groups. The first includes tasks performed during initial acquaintance with identity. They serve as educational material for several consecutive lessons united by one topic. The second group of exercises connects the identity being studied with various applications. This group does not form a compositional unity - the exercises here are scattered on various topics.

The described cycle structure refers to the stage of developing skills in applying specific types of transformations. At the final stage, the synthesis stage, the cycles are modified. Firstly, both groups of tasks are combined to form an “expanded” cycle, and the simplest ones in terms of wording or complexity of completing the task are excluded from the first group. The remaining types of tasks become more complex. Secondly, there is a merging of cycles related to different identities, due to which the role of actions to recognize the applicability of a particular identity increases.

Let us note the features of task cycles related to identities for elementary functions. These features are due to the fact that, firstly, the corresponding identities are studied in connection with the study of functional material and, secondly, they appear later than the identities of the first group and are studied using already formed skills for carrying out identity transformations.

Each newly introduced elementary function dramatically expands the range of numbers that can be designated and named individually. Therefore, the first group of cycle tasks should include tasks to establish connections between these new numerical domains and the original domain of rational numbers. Let's give examples of such tasks.

Example 1. Calculate:

Next to each expression an identity is indicated, in the cycles for which the proposed tasks may be present. The purpose of such tasks is to master the features of records, including symbols of new operations and functions, and to develop mathematical speech skills.

A significant part of the use of identity transformations associated with elementary functions falls on the solution of irrational and transcendental equations. The cycles related to the assimilation of identities include only the simplest equations, but here it is advisable to carry out work on mastering the method of solving such equations: reducing it by replacing the unknown with an algebraic equation.

The sequence of steps for this solution is as follows:

a) find a function for which this equation can be represented in the form;

b) make the substitution and solve the equation;

c) solve each of the equations , where is the set of roots of the equation .

When using the described method, step b) is often performed implicitly, without introducing a notation for . In addition, students often prefer, from the various paths leading to finding an answer, to choose the one that leads to the algebraic equation faster and easier.

Example 2. Solve the equation.

First way:

Second way:

A)

b)

Here you can see that with the first method step a) is more difficult than with the second. The first method is “more difficult to start with,” although the further course of the solution is much simpler. On the other hand, the second method has the advantages of greater ease and greater precision in learning to reduce to an algebraic equation.

For a school algebra course, typical tasks are in which the transition to an algebraic equation is even simpler than in this example. The main load of such tasks relates to the identification of step c) as an independent part of the solution process associated with the use of the properties of the elementary function being studied.

Example 3. Solve the equation:

A) ; b) .

These equations are reduced to the equations: a) or ; b) or . To solve these equations, knowledge of only the simplest facts about the exponential function is required: its monotonicity, range of values. Like the previous example, equations a) and b) can be classified as the first group of a series of exercises for solving quadratic exponential equations.

Thus, we come to a classification of tasks in cycles related to solving transcendental equations that include an exponential function:

1) equations that reduce to equations of the form and have a simple, general answer: ;

2) equations that reduce to equations , where is an integer, or , where ;

3) equations that reduce to equations and require explicit analysis of the form in which the number is written.

Tasks for other elementary functions can be classified similarly.

A significant part of the identities studied in the courses of algebra and algebra and principles of analysis are proved in them or, at least, explained. This aspect of the study of identities is of great importance for both courses, since evidentiary reasoning in them is carried out with the greatest clarity and rigor precisely in relation to identities. Beyond this material, evidence is usually less complete; it is not always distinguished from the substantiation used.

The properties of arithmetic operations are used as the support on which proofs of identities are built.

The educational impact of calculations and identical transformations can be aimed at the development of logical thinking, if only students are systematically required to justify calculations and identical transformations, and at the development of functional thinking, which is achieved in various ways. The importance of calculations and identical transformations in the development of will, memory, intelligence, self-control, and creative initiative is quite obvious.

The demands of everyday and industrial computing practice require students to develop strong, automated skills in rational calculations and identity transformations. These skills are developed in the process of any computational work, however, special training exercises in fast calculations and transformations are necessary.

So, if the lesson involves solving logarithmic equations using the basic logarithmic identity, then it is useful to include in the lesson plan oral exercises on simplifying or calculating the values ​​of expressions: , , . The purpose of the exercises is always communicated to students. During the exercise, it may be necessary to require students to justify individual transformations, actions, or the solution to an entire problem, even if this was not planned. Where different ways of solving a problem are possible, it is advisable to always ask questions: “How was the problem solved?”, “Who solved the problem in a different way?”

The concepts of identity and identity transformation are explicitly introduced in the VI grade algebra course. The very definition of identical expressions cannot be practically used to prove the identity of two expressions, and understand that the essence of identical transformations is to apply to the expression the definitions and properties of those actions that are indicated in the expression, or to add to it an expression that is identically equal to 0, or in multiplying it by an expression identically equal to one. But even having mastered these provisions, students often do not understand why these transformations allow us to assert that the original and resulting expressions are identical, i.e. take the same values ​​for any systems (sets) of variable values.

It is also important to ensure that students clearly understand that such conclusions of identical transformations are consequences of the definitions and properties of the corresponding actions.

The apparatus of identity transformations, accumulated in previous years, is expanded in grade VI. This extension begins with the introduction of an identity expressing the property of the product of powers with the same bases: , where , are integers.

§3. Mathematics program. In the school course “Algebra and the beginnings of analysis,” students systematically study exponential and logarithmic functions and their properties, identical transformations of logarithmic and exponential expressions and their application to solving the corresponding equations and inequalities, and become familiar with basic concepts and statements. In the 11th grade, algebra lessons take 3 hours a week, for a total of 102 hours a year. The program takes 36 hours to study exponential, logarithmic and power functions. The program includes consideration and study of the following issues: The concept of a degree with a rational exponent. Solving irrational equations. Exponential function, its properties and graph. Identical transformations of exponential expressions. Solving exponential equations and inequalities. Logarithm of a number. Basic properties of logarithms. Logarithmic function, its properties and graph. Solving logarithmic equations and inequalities. Derivative of an exponential function. Number and natural logarithm. Derivative of a power function. The main purpose of the exponential and logarithmic function section is to familiarize students with exponential, logarithmic and power functions; teach students to solve exponential and logarithmic equations and inequalities. The concepts of the th root and the degree with a rational exponent are a generalization of the concepts of the square root and the degree with an integer exponent. Students should pay attention to the fact that the properties of roots and powers with rational exponents considered here are similar to those properties possessed by the previously studied square roots and powers with integer exponents. It is necessary to devote enough time to practicing the properties of degrees and developing the skills of identity transformations. The concept of a degree with an irrational exponent is introduced on a visual and intuitive basis. This material plays an auxiliary role and is used when introducing the exponential function. The study of the properties of exponential, logarithmic and power functions is constructed in accordance with the accepted general scheme for studying functions. In this case, an overview of the properties is given depending on the parameter values. Exponential and logarithmic inequalities are solved based on the studied properties of functions. A characteristic feature of the course is the systematization and generalization of students’ knowledge, consolidation and development of skills acquired in the algebra course, which is carried out both when studying new material and when conducting generalized repetition.
Chapter 2. Identical transformations and calculations of exponential and logarithmic expressions

§1. Generalization of the concept of degree.

Definition: The th root of a pure number is a number whose th power is equal to .

According to this definition, the th root of a number is the solution to the equation. The number of roots of this equation depends on and. Let's consider the function. As is known, on the interval this function increases for any value and takes all values ​​from the interval. According to the root theorem, the equation for any has a non-negative root and, moreover, only one. It is called the arithmetic root of the th degree of a number and is denoted by ; the number is called the radical exponent, and the number itself is called the radical expression. The sign is also called a radical.

Definition: The arithmetic root of the th power of a number is a non-negative number whose th power is equal to .

For even numbers the function is even. It follows that if , then the equation, in addition to the root, also has a root. If , then there is one root: ; if , then this equation has no roots, since the even power of any number is non-negative.

For odd values, the function increases along the entire number line; its range is the set of all real numbers. Applying the root theorem, we find that the equation has one root for any and, in particular, for . This root for any value is denoted by .

For roots of odd degree, the equality holds. In fact, , i.e. number is the th root of . But such a root for odd is the only one. Hence, .

Remark 1: For any real

Let us recall the known properties of arithmetic roots of the th degree.

For any natural number, integer and any non-negative integers and the equalities are valid:

1.

2.

3.

4.

Degree with a rational exponent.

The expression is defined for all and except the case at . Let us recall the properties of such powers.

For any numbers , and any integers and the equalities are valid:

We also note that if , then for and for .. and

For students taking the Unified State Exam, mathematics teachers at secondary school No. 26 in Yakutsk use a list of content questions (codifier) ​​for the school mathematics course, the mastery of which is tested when passing the 2007 unified state exam. The elective course in preparation for the Unified State Exam is based on repetition, systematization and deepening of knowledge acquired previously. Classes are held in the form of free...

Example 1 . Calculate:

Next to each expression an identity is indicated, in the cycles for which the proposed tasks may be present. The purpose of such tasks is to master the features of records, including symbols of new operations and functions, and to develop mathematical speech skills.

A significant part of the use of identity transformations associated with elementary functions falls on the solution of irrational and transcendental equations. The cycles related to the assimilation of identities include only the simplest equations, but here it is advisable to carry out work on mastering the method of solving such equations: reducing it by replacing the unknown with an algebraic equation.

The sequence of steps for this solution is as follows:

a) find the function

, for which this equation can be represented as ;

b) make a substitution

and solve the equation ;

c) solve each of the equations

, where is the set of roots of the equation .

When using the described method, step b) is often performed implicitly, without introducing a notation for

. In addition, students often prefer, from the various paths leading to finding an answer, to choose the one that leads to the algebraic equation faster and easier.

Example 2 . Solve the equation

.

First way:

Second way:

Here you can see that with the first method step a) is more difficult than with the second. The first method is “more difficult to start with,” although the further course of the solution is much simpler. On the other hand, the second method has the advantages of greater ease and greater precision in learning to reduce to an algebraic equation.

For a school algebra course, typical tasks are in which the transition to an algebraic equation is even simpler than in this example. The main load of such tasks relates to the identification of step c) as an independent part of the solution process associated with the use of the properties of the elementary function being studied.

Example 3 . Solve the equation:

; b) .

These equations reduce to the equations: a)

or ; b) or . To solve these equations, knowledge of only the simplest facts about the exponential function is required: its monotonicity, range of values. Like the previous example, equations a) and b) can be classified as the first group of a series of exercises for solving quadratic exponential equations.

Thus, we come to a classification of tasks in cycles related to solving transcendental equations that include an exponential function:

1) equations that reduce to equations of the form

and having a simple, general answer: ;

2) equations that reduce to equations

, where is an integer, or , where ;

3) equations that reduce to equations

and requiring explicit analysis of the form in which the number is written .

Tasks for other elementary functions can be classified similarly.

A significant part of the identities studied in the courses of algebra and algebra and principles of analysis are proved in them or, at least, explained. This aspect of the study of identities is of great importance for both courses, since evidentiary reasoning in them is carried out with the greatest clarity and rigor precisely in relation to identities. Beyond this material, evidence is usually less complete; it is not always distinguished from the substantiation used.

The properties of arithmetic operations are used as the support on which proofs of identities are built.

The educational impact of calculations and identical transformations can be aimed at the development of logical thinking, if only students are systematically required to justify calculations and identical transformations, and at the development of functional thinking, which is achieved in various ways. The importance of calculations and identical transformations in the development of will, memory, intelligence, self-control, and creative initiative is quite obvious.

The demands of everyday and industrial computing practice require students to develop strong, automated skills in rational calculations and identity transformations. These skills are developed in the process of any computational work, however, special training exercises in fast calculations and transformations are necessary.

So, if the lesson involves solving logarithmic equations using the basic logarithmic identity

, then it is useful to include in the lesson plan oral exercises on simplifying or calculating the meanings of expressions: , , . The purpose of the exercises is always communicated to students. During the exercise, it may be necessary to require students to justify individual transformations, actions, or the solution to an entire problem, even if this was not planned. Where different ways of solving a problem are possible, it is advisable to always ask questions: “How was the problem solved?”, “Who solved the problem in a different way?”

The concepts of identity and identity transformation are explicitly introduced in the VI grade algebra course. The very definition of identical expressions cannot be practically used to prove the identity of two expressions, and understand that the essence of identical transformations is to apply to the expression the definitions and properties of those actions that are indicated in the expression, or to add to it an expression that is identically equal to 0, or in multiplying it by an expression identically equal to one. But even having mastered these provisions, students often do not understand why these transformations allow us to assert that the original and resulting expressions are identical, i.e. take the same values ​​for any systems (sets) of variable values.

It is also important to ensure that students clearly understand that such conclusions of identical transformations are consequences of the definitions and properties of the corresponding actions.

The apparatus of identity transformations, accumulated in previous years, is expanded in grade VI. This extension begins by introducing an identity expressing the property of the product of powers with the same bases:

The concept of a degree with a rational exponent. Solving irrational equations. Exponential function, its properties and graph. Identical transformations of exponential expressions. Solving exponential equations and inequalities. Logarithm of a number. Basic properties of logarithms. Logarithmic function, its properties and graph. Solving logarithmic equations and inequalities. Derivative of an exponential function. Number and natural logarithm. Derivative of a power function.

The main purpose of the exponential and logarithmic function section is to familiarize students with exponential, logarithmic and power functions; teach students to solve exponential and logarithmic equations and inequalities.

The concepts of the th root and the degree with a rational exponent are a generalization of the concepts of the square root and the degree with an integer exponent. Students should pay attention to the fact that the properties of roots and powers with rational exponents considered here are similar to those properties possessed by the previously studied square roots and powers with integer exponents. It is necessary to devote enough time to practicing the properties of degrees and developing the skills of identity transformations. The concept of a degree with an irrational exponent is introduced on a visual and intuitive basis. This material plays an auxiliary role and is used when introducing the exponential function.

The study of the properties of exponential, logarithmic and power functions is constructed in accordance with the accepted general scheme for studying functions. In this case, an overview of the properties is given depending on the parameter values. Exponential and logarithmic inequalities are solved based on the studied properties of functions.

A characteristic feature of the course is the systematization and generalization of students’ knowledge, consolidation and development of skills acquired in the algebra course, which is carried out both when studying new material and when conducting generalized repetition.