Independent work logarithmic inequalities. Verification work "solution of logarithmic equations". Used methods and techniques
Sections: Maths
Logarithmic equations, inequalities and systems of logarithmic inequalities are among the tasks offered at the unified state exam in mathematics. The manual can be used to prepare for the unified state exam, as well as for a deeper study of the topic “Logarithmic function. Solution of logarithmic equations, inequalities and systems of logarithmic inequalities”.
This manual presents independent work for practicing and consolidating the skills of solving logarithmic equations, inequalities and systems of logarithmic inequalities.
Independent work is designed for students of physical and mathematical classes, however, it can also be used for well-performing students of educational institutions. For each of the work performed, an assessment 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 an independent work in which students are asked to solve logarithmic equations, using the definition of the logarithm, the basic logarithmic identity and other transformations of logarithms. In the process of solving, 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 process of solving 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 the trigonometric inequality and select the necessary roots in accordance with the obtained restriction. Tasks 3 and 4 are the most difficult to work with and are designed for a higher level of student preparation. It is useful to use this work in a secondary school for better memorization and assimilation of the basic concepts on this topic, excluding tasks 3 and 4 from it.
Appendix 2 contains independent work on solving logarithmic inequalities. Various types of logarithmic inequalities are included in the work. At the same time, it is advisable to give tasks 1, 2 and 3 to students of a comprehensive school. To solve inequality 4, students will need skills in working with inequalities containing a module. Inequalities 4, 5 and 6 are intended for students of physical and mathematical classes.
Appendix 3 contains three systems of inequalities, each of which contains a logarithmic inequality with a variable in the base, as well as an exponential inequality, reduced to a square one using a change of variable, or solved using the generalized interval method. This independent work is designed for students with a fairly high level of mathematical training and is recommended for classes with in-depth study of mathematics.
Independent work is compiled in four variants of equivalent complexity, which are convenient to use for intermediate control of students' knowledge, development of practical skills in solving problems on the topic “Logarithmic function”.
The works presented in the manual allow students to better master the material covered on the specified topic, which is confirmed by practice.
Independent work contains answers, which will significantly reduce the time for checking the work by the teacher.
This manual can also be used to organize repetition in preparing high school students for the successful passing of the unified state exam in mathematics.
Literature
- Tsypkin A.G., Pinsky A.I. Reference manual in mathematics with methods for solving problems for applicants to universities - M .: "Oniks Publishing House", 2007.
- Sergeev I.N., Panferov V.S. USE 2013. Mathematics. Task C3. Equations and inequalities - Moscow: "MTsNMO Publishing House", 2013.
- Kolesnikova S.I. Exponential and logarithmic equations. USE. Maths. - Moscow: LLC "Azbuka - 2000", 2012.
- Kolesnikova S.I. Exponential and logarithmic inequalities. USE. Maths. - Moscow: Azbuka - 2000 LLC, 2013.
- Yashchenko I.V., Shestakov S.A., Trepalin A.S., Zakharov P.I. Preparing for the Unified State Examination in Mathematics. New demo version 2014.- Moscow: “MTsNMO Publishing House”, 2014.
Used Internet resources
- http://reshuege.ru/
MBOU secondary school №92 Kemerovo
Test work in mathematics.
Topic: "Solution of logarithmic equations". Tasks B5 from the open bank of USE tasks (http://mathege.ru/)
Prepared by: math teacher
MBOU secondary school №92 Kemerovo
Denisova Tatyana Alexandrovna
Task B5 in the exam tests the ability to solve simple equations. This development is devoted to one of the sections of task B5 - this is the solution of logarithmic equations.
The main task is:
Checking the quality of knowledge and skills of students;
Increasing the Computing Culture of Students
The presented test work consists of 4 options, each of which has 13 tasks. The tasks of this work correspond to the prototypes of tasks B5 from the open bank of USE tasks in mathematics. This material can be used in preparation for the exam. For ease of verification, the answers are given.
Test on logarithmic equations, tasks B5 from the open bank of USE tasks option 1
Test on logarithmic equations, tasks B5 from the open bank of USE tasks option 2
Test on logarithmic equations, tasks B5 from the open bank of USE tasks option 3.
Test on logarithmic equations, tasks B5 from the open bank of USE tasks option 4
Answers to the test work
| 1 option | |||||||||||||
| Option 2 | |||||||||||||
| 3 option | |||||||||||||
| 4 option |
Class: 11
Lesson type: iterative-generalizing
Lesson Objectives:
- educational: to generalize 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 mindfulness, analytical thinking, the ability to exercise self- and mutual control;
- educational: education of positive motivation for learning, culture of mathematical speech.
Used methods and techniques:
- explanatory and illustrative,
- reproductive,
- method of control and correction of knowledge
Forms of work:
- frontal,
- pair work,
- individual
Equipment: interactive whiteboard, computer, projector
During the classes
Lesson stage |
Teacher activity | Student activities |
| Org.moment | Greetings | Welcome teacher |
| Statement of the learning task | - Guys, the topic of today's lesson is “Inequalities containing logarithmic expressions”. Try to formulate the goals and objectives of the lesson on your own. | Write down the topic of the lesson. They independently formulate the goals and objectives of the lesson. |
| Update | - Remember and formulate the definition of the logarithm, the properties of logarithms. What is a logarithmic function? List the properties of the logarithmic function, 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 = log0.2x; 4) y = log0.5 (2x+5); 5) y = log3 (x+2) Using the properties of a 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. |
Orally perform the task | |
| Mathematical dictation
|
Perform a mathematical dictation with further self-examination and error correction | |
| Repetition, generalization and systematization of the studied material | Logarithmic inequalities An inequality containing a variable only under the sign of the logarithm is called a logarithmic one.
Example 1. Solve an inequality
Example 2. Solve the inequality
Among standard inequalities, a special place is occupied by logarithmic inequalities containing a variable at the base of the logarithm, since the solution of 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 rationalization of inequalities: Example 4. Solve the inequality The sign of the difference is the same as the sign of the difference provided that x Answer: x |
Listen to the teacher's explanation, make the necessary notes in notebooks Solution with comments |
| Application of acquired knowledge | Solve the inequalities:
|
Three students simultaneously solve on the board, the rest - in notebooks, then check their solutions |
| Independent work | Option 1.
Option 2.
|
Perform independent work |
| D / s | №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 in doing your own work? |
They reflect on their own activities. |
Literature.
- Mordkovich A.G. Algebra and the beginnings of analysis. Grade 11. At 2 pm Part 1. Textbook for educational institutions (profile level) / A.G. Mordkovich, P.V. Semyonov. – M.: Mnemozina, 2012. – 287 p.: ill.
- Cherkasov O.Yu., Yakushev A.G. Mathematics: Intensive Exam Preparation Course. – 7th ed. – M.: Iris-press, 2003.-432p.: ill. – (Home tutor).
