Class: 11

Lesson type: repeating and generalizing

Lesson objectives:

• educational: summarize and systematize students’ knowledge on the topic “Logarithmic inequalities”, consider non-standard methods for solving logarithmic inequalities, check the level of students’ knowledge on the topic of the lesson;
• developing: development of attentiveness, analytical thinking, ability to exercise self- and mutual control;
• educational: nurturing positive motivation for learning, a culture of mathematical speech.

Methods and techniques used:

• explanatory and illustrative,
• reproductive,
• method of control and correction of knowledge

Forms of work:

• frontal,
• work in pairs,
• individual

Equipment: interactive whiteboard, computer, projector

During the classes

Lesson stage	Teacher's activities	Student activities
Organizational moment	Greetings	Greet the teacher
Setting a learning task	- Guys, the topic of today’s lesson is “Inequalities containing logarithmic expressions.” Try to formulate the goals and objectives of the lesson yourself.	Write down the topic of the lesson. They independently formulate the goals and objectives of the lesson.
Update	- Remember and formulate the definition of a logarithm, the properties of logarithms. Which function is called logarithmic? List the properties of the logarithmic function and schematically draw its graph. Which logarithmic function is increasing (decreasing)?	Answer the teacher's questions
	- Determine which of the following functions are increasing and which are decreasing: 3) y = log0.2 x; 4) y = log0.5 (2x+5); 5) y = log3 (x+2) Using the properties of the logarithmic function, compare: a) log2 3 and log2 5; b) log2 1/3 and log2 1/5; c)log1/2 3 and log1/2 5; d)log1/2 1/3 and log1/2 1/5.	Perform the task orally
	Mathematical dictation	Perform a mathematical dictation with further self-testing and error correction
Repetition, generalization and systematization of the studied material	Logarithmic inequalities An inequality containing a variable only under the logarithmic sign is called logarithmic. Example 1. Solve inequality Example 2: Solve the inequality Among standard inequalities, a special place is occupied by logarithmic inequalities containing a variable in the base of the logarithm, since solving such inequalities causes certain difficulties. The most common way to solve such inequalities is to consider the cases: 1) the base is greater than 1; 2) the base is positive and less than 1. Example 3: Solve the inequality It is more convenient to solve inequalities of this type using the method of rationalizing inequalities: Example 4: Solve the inequality The sign of the difference coincides with the sign of the difference provided that x Answer: x	Listen to the teacher’s explanation and make the necessary notes in their notebooks. Solution with comments
Application of acquired knowledge	Solve the inequalities:	Three students solve simultaneously on the board, the rest - in notebooks, then compare their solutions
Independent work	Option 1. Option 2.	Perform independent work
D/z	№28.16, 28.47, 30.43	Write down homework
Lesson summary	- Have we completed the tasks set at the beginning of the lesson? What difficulties did you encounter when doing independent work?	Carry out reflection on their own activities.

Literature.

1. Mordkovich A.G. Algebra and the beginnings of analysis. Grade 11. In 2 hours. Part 1. Textbook for general education institutions (profile level) / A.G. Mordkovich, P.V. Semenov. – M.: Mnemosyne, 2012. – 287 p.: ill.
2. Cherkasov O.Yu., Yakushev A.G. Mathematics: intensive exam preparation course. – 7th ed. – M.: Iris-press, 2003.-432 p.: ill. – (Home tutor).

Sections: Mathematics

Logarithmic equations, inequalities, and systems of logarithmic inequalities are among the problems offered on the Unified State Mathematics Exam. The manual can be used to prepare for the unified state exam, as well as for a more in-depth study of the topic “Logarithmic function. Solving logarithmic equations, inequalities and systems of logarithmic inequalities.”

This manual presents independent work to practice and consolidate skills in solving logarithmic equations, inequalities and systems of logarithmic inequalities.

Independent work is designed for students in physics and mathematics classes, however, they can also be used for well-performing students in general education institutions. For each of the completed works, a grade is given, which will serve as sufficient motivation for the most complete and high-quality home study of the material covered the day before.

Appendix 1 contains independent work in which students are asked to solve logarithmic equations using the definition of a logarithm, the basic logarithmic identity and other transformations of logarithms. During the solution process, it is necessary to check the received answers for compliance with the restrictions provided for when using the logarithmic function. In addition, one of the logarithmic equations in the solution process will require trigonometric transformations, as well as checking the found roots for compliance with the restrictions introduced in connection with the use of the logarithm, i.e. Students will have to solve trigonometric inequalities and select the necessary roots in accordance with the resulting constraint. Tasks 3 and 4 are the most difficult to work with and are designed for a higher level of student preparation. It is also useful to use this work in secondary school for better memorization and assimilation of basic concepts on this topic, excluding tasks 3 and 4 from it.

Appendix 2 contains independent work on solving logarithmic inequalities. The work includes various types of logarithmic inequalities. In this case, it is advisable to give tasks 1, 2 and 3 to secondary school students. Solving Inequality 4 will require students to have skills in working with inequalities contained in the module. Inequalities 4, 5 and 6 are intended for students in physics and mathematics classes.

Appendix 3 shows three systems of inequalities, each of which contains a logarithmic inequality with a variable in the base, as well as an exponential inequality that can be reduced to a quadratic inequality using a change of variable, or solved using the generalized method of intervals. This independent work is designed for students with a fairly high level of mathematical preparation and is recommended for use in classes with in-depth study of mathematics.

Independent works are compiled in four variants of equivalent complexity, which are convenient to use for intermediate control of students’ knowledge and development of practical skills in solving problems on the topic “Logarithmic function”.

The work presented in the manual allows students to better understand the material covered on this topic, which is confirmed by practice.

Independent work contains answers, which will significantly reduce the time the teacher checks the work.

This manual can also be used to organize repetition when preparing high school students to successfully pass the unified state exam in mathematics.

Literature

1. Tsypkin A.G., Pinsky A.I. A reference manual on mathematics with methods for solving problems for those entering universities - M.: “Onyx Publishing House”, 2007.
2. Sergeev I.N., Panferov V.S. Unified State Exam 2013. Mathematics. Task C3. Equations and inequalities - Moscow: “MCNMO Publishing House”, 2013.
3. Kolesnikova S.I. Exponential and logarithmic equations. Unified State Exam. Mathematics. – Moscow: LLC “Azbuka – 2000”, 2012.
4. Kolesnikova S.I. Exponential and logarithmic inequalities. Unified State Exam. Mathematics. – Moscow: LLC “Azbuka – 2000”, 2013.
5. Yashchenko I.V., Shestakov S.A., Trepalin A.S., Zakharov P.I. Preparation for the Unified State Exam in mathematics. New demo version 2014. - Moscow: “MCNMO Publishing House”, 2014.

Internet resources used

1. http://reshuege.ru/

MBOU Secondary School No. 92, Kemerovo

Mathematics test.

Topic: “Solving logarithmic equations.” B5 tasks from the open Unified State Exam task bank (http://mathege.ru/)

Prepared by: math teacher

MBOU Secondary School No. 92, Kemerovo

Denisova Tatyana Alexandrovna

Task B5 in the Unified State Exam tests the ability to solve simple equations. This development is devoted to one of the sections of task B5 - solving logarithmic equations.

The main task is:

Checking the quality of students' knowledge and skills;

Improving the computing culture of students

The presented test work consists of 4 options, each of which has 13 tasks. The tasks in this work correspond to the prototypes of B5 tasks from the open bank of Unified State Examination tasks in mathematics. This material can be used in preparation for the Unified State Exam. For ease of verification, the answers are given.

Test on logarithmic equations, tasks B5 from the open Unified State Exam task bank option 1

Test on logarithmic equations, tasks B5 from the open bank of Unified State Examination tasks option 2

Test on logarithmic equations, tasks B5 from the open bank of Unified State Examination tasks option 3.

Test on logarithmic equations, tasks B5 from the open Unified State Exam task bank option 4

Answers to the test work

1 option

Option 2

Option 3

Option 4