Real logarithm
Logarithm of a real number log a b makes sense with src="/pictures/wiki/files/55/7cd1159e49fee8eff61027c9cde84a53.png" border="0">.
The most widely used are the following types of logarithms.
If we consider a logarithmic number as a variable, we get logarithmic function, For example: . This function is defined on the right side of the number line: x> 0 , is continuous and differentiable there (see Fig. 1).
Properties
natural logarithms
For , the equality
| (1) |
In particular,
This series converges faster, and in addition, the left side of the formula can now express the logarithm of any positive number.
Relationship with the decimal logarithm: .
Decimal logarithms
Rice. 2. Log scale
Logarithms to base 10 (symbol: lg a) before the invention of calculators were widely used for calculations. The non-uniform scale of decimal logarithms is commonly applied to slide rules as well. A similar scale is widely used in various fields of science, for example:
- Chemistry - the activity of hydrogen ions ().
- Music theory - the musical scale, in relation to the frequencies of musical sounds.
The logarithmic scale is also widely used to identify the exponent in exponential dependences and the coefficient in the exponent. At the same time, a graph plotted on a logarithmic scale along one or two axes takes the form of a straight line, which is easier to study.
Complex logarithm
Multivalued function
Riemann surface
The complex logarithmic function is an example of a Riemann surface; its imaginary part (Fig. 3) consists of an infinite number of branches twisted like a spiral. This surface is simply connected; its only zero (of the first order) is obtained by z= 1 , special points: z= 0 and (branch points of infinite order).
The Riemann surface of the logarithm is the universal covering for the complex plane without the point 0 .
Historical outline
Real logarithm
The need for complex calculations in the 16th century grew rapidly, and much of the difficulty was associated with multiplication and division of multi-digit numbers. At the end of the century, several mathematicians, almost simultaneously, came up with the idea: to replace time-consuming multiplication with simple addition, comparing geometric and arithmetic progressions using special tables, while the geometric one will be the original one. Then the division is automatically replaced by an immeasurably simpler and more reliable subtraction. He was the first to publish this idea in his book Arithmetica integra»Michael Stiefel, who, however, did not make serious efforts to implement his idea.
In the 1620s, Edmund Wingate and William Oughtred invented the first slide rule, before the advent of pocket calculators, an indispensable tool for an engineer.
A close to modern understanding of logarithm - as an operation inverse to exponentiation - first appeared in Wallis and Johann Bernoulli, and was finally legalized by Euler in the 18th century. In the book "Introduction to the Analysis of Infinite" (), Euler gave modern definitions of both exponential and logarithmic functions, expanded them into power series, and especially noted the role of the natural logarithm.
Euler also has the merit of extending the logarithmic function to the complex domain.
Complex logarithm
The first attempts to extend logarithms to complex numbers were made at the turn of the 17th-18th centuries by Leibniz and Johann Bernoulli, but they failed to create a holistic theory - primarily for the reason that the very concept of the logarithm was not yet clearly defined. The discussion on this issue was first between Leibniz and Bernoulli, and in the middle of the XVIII century - between d'Alembert and Euler. Bernoulli and d'Alembert believed that it was necessary to define log(-x) = log(x). The complete theory of the logarithms of negative and complex numbers was published by Euler in 1747-1751 and is essentially no different from the modern one.
Although the dispute continued (D'Alembert defended his point of view and argued it in detail in an article in his Encyclopedia and in other works), Euler's point of view quickly gained universal recognition.
Logarithmic tables
Logarithmic tables
It follows from the properties of the logarithm that instead of the time-consuming multiplication of multi-digit numbers, it is enough to find (according to the tables) and add their logarithms, and then perform potentiation using the same tables, that is, find the value of the result by its logarithm. Doing division differs only in that logarithms are subtracted. Laplace said that the invention of logarithms "prolonged the life of astronomers" by greatly speeding up the process of calculation.
When moving the decimal point in a number to n digits, the value of the decimal logarithm of this number is changed by n. For example, lg8314.63 = lg8.31463 + 3 . It follows that it is enough to make a table of decimal logarithms for numbers in the range from 1 to 10.
The first tables of logarithms were published by John Napier (), and they contained only the logarithms of trigonometric functions, and with errors. Independently of him, Jost Burgi, a friend of Kepler, published his tables (). In 1617 Oxford mathematics professor Henry Briggs published tables that already included the decimal logarithms of the numbers themselves, from 1 to 1000, with 8 (later 14) digits. But there were also errors in the Briggs tables. The first error-free edition based on the Vega tables () appeared only in 1857 in Berlin (Bremiver tables).
In Russia, the first tables of logarithms were published in 1703 with the participation of L. F. Magnitsky. Several collections of tables of logarithms were published in the USSR.
- Bradis V. M. Four-digit mathematical tables. 44th edition, M., 1973.
Algebra lesson in 10th grade
Topic: "Logarithmic function, its properties and graph"
Goals:
educational: Introduce the concept of a logarithmic function using past experience, give a definition. Learn the basic properties of the logarithmic function. To form the ability to perform the construction of a graph of a logarithmic function.
Developing: Develop the ability to highlight the main thing, compare, generalize. To form a graphic culture of students.
Educational: Show the relationship of mathematics with the surrounding reality. To form communication skills, dialogue, the ability to work in a team.
Lesson type: Combined
Teaching methods: Partial search, dialogue.
During the classes.
1. Actualization of past experience:
Students are offered oral exercises using the definition of the logarithm, its properties, formulas for the transition to a new base, solving the simplest logarithmic and exponential equations, examples for finding the range of acceptable values for logarithmic expressions
oral exercisesoral work.
1) Calculate using the definition of the logarithm: log 2 8; log 4 16;.
2) Calculate using the basic logarithmic identity:
3) Solve the equation using the definition:
4) Find out for what values of x the expression makes sense:
5) Find the value of the expression using the properties of logarithms:
2. Studying the topic. Students are invited to solve exponential equations: 2 x \u003d y; () x = y. by expressing x in terms of y. As a result of this work, formulas are obtained that define functions unfamiliar to students. ,. Question : "What would you call this feature?" students say that it is logarithmic, since the variable is under the sign of the logarithm:.
Question . Define a function. Definition: The function defined by the formula y=log a x is called logarithmic with base a (a>0, and 1)
III. Function research y=log a x
More recently, we have introduced the notion of the logarithm of a positive number with respect to base a, which is positive and different from 1. For any positive number, you can find the logarithm in a given base. But then you should also think about a function like y=log ax, and about its graphics and properties.The function given by the formula y=log a x is called logarithmic with base a (a>0, and 1)
The main properties of the logarithmic function:
1. The domain of definition of the logarithmic function will be the entire set of positive real numbers. For brevity, it is also referred to asR+. An obvious property, since every positive number has a logarithm to the base a.D(f)=R+
2. The area of value of the logarithmic function will be the entire set of real numbers.E(f)= (-∞; +∞)
3 . The graph of a logarithmic function always passes through the point (1; 0).
4 . Llogarithmic function of ageem at a>1, and decreases at 0<х<1.
5 . The function is neither even nor odd. Logarithmic function - general form functionA.
6 . The function has no maximum and minimum points, is continuous in the domain of definition.
The following figure is a graph of a decreasing logarithmic function - (0
If you build exponential and logarithmic functions with the same bases in the same coordinate axis, then the graphs of these functions will be symmetrical with respect to the straight line y \u003d x. This statement is shown in the following figure.
The above statement will be true for both increasing and decreasing logarithmic and exponential functions.
Consider an example: find the domain of the logarithmic function f(x) = log 8 (4 - 5x).
Based on the properties of the logarithmic function, the domain of definition is the entire set of positive real numbers R+. Then the given function will be defined for such x for which 4 - 5x>0. We solve this inequality and get x<0.8. Таким образом, получается, что областью определения функции f(x) = log 8 (4 - 5*x) will be the interval (-∞;0.8)
Graphs of the logarithmic function in the GeoGebra program
Graphs of the logarithmic function
1) natural logarithm y = ln (x)
2) decimal logarithm y = lg (x)
3) base 2 logarithm y = ld (x)
V. Fixing the topic
Applying the obtained properties of the logarithmic function, we will solve the following tasks:
1. Find the domain of the function: y=log 8 (4-5x); y=log 0.5 (2x+8);.
3. Schematically construct graphs of functions: y \u003d log 2 (x + 2) -3 y \u003d log 2 (x) +2
Ministry of Education and Youth Policy of the Chuvash Republic
State Autonomous Professional
educational institution of the Chuvash Republic
"Cheboksary College of Transport and Construction Technologies"
(GAPOU "Cheboksary Technical School TransStroyTekh"
Ministry of Education of Chuvashia)
Methodical development
ODP. 01 Mathematics
"Logarithmic function. Properties and Graph»
Cheboksary - 2016
Explanatory note………………………………………………………………………………………………. ......…………………………………….….…3
Theoretical substantiation and methodical implementation…………….….................................4-10
Conclusion…………………………………………………………….......................... .........................………....eleven
Applications……………………………………………………………………………………………….. ..........................………...13
Explanatory note
Methodological development of the lesson module in the discipline "Mathematics" on the topic "Logarithmic function. Properties and Graph” from the section “Roots, Degrees and Logarithms” is compiled on the basis of the Work Program in Mathematics and the calendar-thematic plan. The topics of the lesson are interconnected by the content, main provisions.
The purpose of studying this topic is to learn the concept of a logarithmic function, to study its basic properties, to learn how to plot a logarithmic function and learn to see a logarithmic spiral in the world around us.
The program material of this lesson is based on the knowledge of mathematics. The methodological development of the lesson module was compiled for conducting theoretical classes on the topic: “Logarithmic function. Properties and Graph” -1 hour. During the practical lesson, students consolidate their knowledge: definitions of functions, their properties and graphs, graph transformations, continuous and periodic functions, inverse functions and their graphs, logarithmic functions.
Methodological development is intended to provide methodological assistance to students in the study of the lesson module on the topic “Logarithmic function. Properties and Graph. As an extracurricular independent work, students can prepare a message on the topic “Logarithms and their application in nature and technology”, crossword puzzles and rebuses using additional sources. Educational knowledge and professional competencies obtained in the study of the topic "Logarithmic functions, their properties and graphs" will be applied in the study of the following sections: "Equations and inequalities" and "Beginnings of mathematical analysis".
Didactic lesson structure:
Subject:« Logarithmic function. Properties and Graph »
Lesson type: Combined.
Lesson objectives:
Educational- the formation of knowledge in the assimilation of the concept of a logarithmic function, the properties of a logarithmic function; use graphs to solve problems.
Educational- the development of mental operations through concretization, the development of visual memory, the need for self-education, to promote the development of cognitive processes.
Educational- education of cognitive activity, sense of responsibility, respect for each other, mutual understanding, self-confidence; fostering a culture of communication; fostering a conscious attitude and interest in learning.
Means of education:
Methodological development on the topic;
Personal Computer;
Textbook Sh.A Alimov "Algebra and the beginning of analysis" grade 10-11. Publishing house "Enlightenment".
Internal connections: exponential function and logarithmic function.
Interdisciplinary connections: algebra and mathematical analysis.
Studentmust know:
definition of a logarithmic function;
properties of the logarithmic function;
graph of a logarithmic function.
Studentshould be able to:
perform transformations of expressions containing logarithms;
find the logarithm of a number, apply the properties of logarithms when taking a logarithm;
determine the position of a point on the graph by its coordinates and vice versa;
apply the properties of the logarithmic function when plotting graphs;
Perform chart transformations.
Lesson plan
1. Organizational moment (1 min).
2. Setting the goal and objectives of the lesson. Motivation of educational activity of students (1 min).
3. The stage of updating the basic knowledge and skills (3 min).
4. Checking homework (2 min).
5. Stage of assimilation of new knowledge (10 min).
6. Stage of consolidation of new knowledge (15 min).
7. Control of the material learned in the lesson (10 min).
8. Summing up (2 min).
9. The stage of informing students about homework (1 min).
During the classes:
1. Organizational moment.
Includes a greeting by the teacher of the class, preparation of the room for the lesson, checking absentees.
2. Setting goals and objectives of the lesson.
Today we will talk about the concept of a logarithmic function, draw a graph of a function, and study its properties.
3. The stage of updating basic knowledge and skills.
It is carried out in the form of frontal work with the class.
What was the last function we studied? Sketch it on the board.
Define an exponential function.
What is the root of the exponential equation?
What is the definition of a logarithm?
What are the properties of logarithms?
What is the basic logarithmic identity?
4. Checking homework.
Students open notebooks and show the solved exercises. Ask questions that come up while doing homework.
5. The stage of assimilation of new knowledge.
Teacher: Open notebooks, write down today's date and the topic of the lesson "Logarithmic function, its properties and graph."
Definition: A logarithmic function is a function of the form
Where is a given number, .
Consider the construction of a graph of this function using a specific example.
We construct graphs of functions and .
| |
|
| |
|
Note 1: The logarithmic function is the inverse of the exponential function, where 
. Therefore, their graphs are symmetrical with respect to the bisector of the I and III coordinate angles (Fig. 1).
Based on the definition of the logarithm and the type of graphs, we reveal the properties of the logarithmic function:
1) Domain of definition: , because by definition of the logarithm x>0.
2) Range of function values: .
3) The logarithm of the unit is equal to zero, the logarithm of the base is equal to one: , .
4) The function , increases in the interval (Fig. 1).
5) The function , decrease in the interval (Fig. 1).
6) Intervals of sign constancy:
If , then at ; at ;
If , then at at ;
Note 2: The graph of any logarithmic function always passes through the point (1; 0).
Theorem: If 
, where , then .
6. Stage of consolidation of new knowledge.
Teacher: We solve tasks No. 318 - No. 322 (odd) (§18Alimov Sh.A. “Algebra and the beginning of analysis”, grade 10-11).
1) 
because the function is increasing.
3) , because the function is decreasing.
1) , because and .
3) , because and .
1) , since , , then .
3) , because 10> 1, , then .
1) decreasing
3) is increasing.
7. Summing up.
- Today we did a good job at the lesson! What new did you learn at the lesson today?
(New kind of function - logarithmic function)
Formulate the definition of a logarithmic function.
(The function y = logax, (a > 0, a ≠ 1) is called the logarithmic function)
Well done! Right! Name the properties of the logarithmic function.
(domain of a function, set of values of a function, monotonicity, constancy)
8. Control of the material learned in the lesson.
Teacher: Let's find out how well you have learned the topic “Logarithmic function. Properties and Graph. To do this, we will write a test paper (Appendix 1). The work consists of four tasks that must be solved using the properties of the logarithmic function. You have 10 minutes to complete the test.
9. The stage of informing students about homework.
Writing on the board and in the diaries: Alimov Sh.A. "Algebra and the beginning of analysis" 10-11 grade. §18 #318 - #322 (even)
Conclusion
In the course of using the methodological development, we have achieved all the goals and objectives set. In this methodological development, all the properties of the logarithmic function were considered, thanks to which students learned to perform transformations of expressions containing logarithms and build graphs of logarithmic functions. The implementation of practical tasks helps to consolidate the studied material, and the control of testing knowledge and skills will help teachers and students find out how effective their work was in the lesson. Methodological development allows students to obtain interesting and informative information on the topic, generalize and systematize knowledge, apply the properties of logarithms and the logarithmic function when solving various logarithmic equations and inequalities.
Alimov Sh.A., Kolyagin Yu.M., Sidorov Yu.V., Fedorova N.E., Shabunin M.I. - M. Education, 2011.
Nikolsky S. M., Potapov M. K., Reshetnikov N. N. et al. Algebra and the beginnings of mathematical analysis (basic and profile levels). 10 cells - M., 2006.
Kolyagin Yu.M., Tkacheva M.V., Federova N.E. and others, ed. Zhizhchenko A.B. Algebra and the beginnings of mathematical analysis (basic and profile levels). 10 cells - M., 2005.
Lisichkin V. T. Mathematics in problems with solutions: textbook / V. T. Lisichkin, I. L. Soloveychik. - 3rd ed., erased. - St. Petersburg. [and others] : Lan, 2011 (Arkhangelsk). - 464 p.
Internet resources:
http://school- collection.edu.ru - Electronic textbook "Mathematics in
school, 21st century.
http://fcior.edu.ru - information, training and control materials.
www.school-collection.edu.ru - Unified collection of Digital educational resources.
Applications
Option 1.
Option 2.
Criteria for evaluation:
Mark "3" (satisfactory) is placed for any 2 correctly executed examples.
The mark "4" (good) is given if any 3 examples are correctly performed.
The mark "5" (excellent) is placed for all 4 correctly executed examples.
The concept of a logarithmic function
First, let's remember what a logarithm is.
Definition 1
The logarithm of a number $b\in R$ to the base $a$ ($a>0,\ a\ne 1$) is the number $c$ to which the number $a$ must be raised to obtain the number $b$.
Consider the exponential function $f\left(x\right)=a^x$, where $a >1$. This function is increasing, continuous and maps the real axis to the interval $(0,+\infty)$. Then, by the theorem on the existence of an inverse continuous function, in the set $Y=(0,+\infty)$ it has an inverse function $x=f^(-1)(y)$, which is also continuous and increases in $Y $ and maps the interval $(0,+\infty)$ to the entire real axis. This inverse function is called the logarithmic function in base $a\ (a >1)$ and is denoted $y=((log)_a x\ )$.
Now consider the exponential function $f\left(x\right)=a^x$, where $0
Thus, we have defined a logarithmic function for all possible values of the base $a$. Let us consider these two cases separately.
1%24"> Function $y=((log)_a x\ ),\ a >1$
Consider properties this function.
There are no intersections with the $Oy$ axis.
The function is positive for $x\in (1,+\infty)$ and negative for $x\in (0,1)$
$y"=\frac(1)(xlna)$;
Minimum and maximum points:
The function increases over the entire domain of definition;
$y^("")=-\frac(1)(x^2lna)$;
\[-\frac(1)(x^2lna)The function is convex on the entire domain of definition;
$(\mathop(lim)_(x\to 0) y\ )=-\infty ,\ (\mathop(lim)_(x\to +\infty ) y\ )=+\infty ,\ $;
Function graph (Fig. 1).
Figure 1. Graph of the function $y=((log)_a x\ ),\ a >1$
Function $y=((log)_a x\ ), \ 0
Consider the properties of this function.
The domain of definition is the interval $(0,+\infty)$;
The range of value is all real numbers;
The function is neither even nor odd.
Intersection points with coordinate axes:
There are no intersections with the $Oy$ axis.
For $y=0$, $((log)_a x\ )=0,\ x=1.$ Intersection with the $Ox$ axis: (1,0).
The function is positive for $x\in (0,1)$ and negative for $x\in (1,+\infty)$
$y"=\frac(1)(xlna)$;
Minimum and maximum points:
\[\frac(1)(xlna)=0-roots\ no\]
There are no maximum or minimum points.
$y^("")=-\frac(1)(x^2lna)$;
Convexity and concavity intervals:
\[-\frac(1)(x^2lna)>0\]
Function graph (Fig. 2).
Examples of research and construction of logarithmic functions
Example 1
Explore and graph the function $y=2-((log)_2 x\ )$
The domain of definition is the interval $(0,+\infty)$;
The range of value is all real numbers;
The function is neither even nor odd.
Intersection points with coordinate axes:
There are no intersections with the $Oy$ axis.
For $y=0$, $2-((log)_2 x\ )=0,\ x=4.$ Intersection with the $Ox$ axis: (4,0).
The function is positive for $x\in (0,4)$ and negative for $x\in (4,+\infty)$
$y"=-\frac(1)(xln2)$;
Minimum and maximum points:
\[-\frac(1)(xln2)=0-roots\ no\]
There are no maximum or minimum points.
The function decreases over the entire domain of definition;
$y^("")=\frac(1)(x^2ln2)$;
Convexity and concavity intervals:
\[\frac(1)(x^2ln2) >0\]
The function is concave over the entire domain of definition;
$(\mathop(lim)_(x\to 0) y\ )=+\infty ,\ (\mathop(lim)_(x\to +\infty ) y\ )=-\infty ,\ $;
Figure 3
Lesson type: learning new material.
Lesson Objectives:
- form a representation of the logarithmic function, its basic properties;
- to form the ability to plot a graph of a logarithmic function;
- to promote the development of skills to identify the properties of the logarithmic function according to the schedule;
- development of skills in working with text, the ability to analyze information, the ability to systematize, evaluate, use it;
- development of skills to work in pairs, microgroups (communication skills, dialogue, making a joint decision)
Technology used: technology for the development of critical thinking, technology for working in collaboration
Techniques used: true, false statements, INSERT, cluster, cinquain
The lesson uses elements of technology for the development of critical thinking to develop the ability to identify gaps in one’s knowledge and skills when solving a new problem, assess the need for this or that information for one’s activity, carry out information retrieval, independently master the knowledge necessary to solve cognitive and communicative tasks. This type of thinking helps to be critical of any statements, not to take anything for granted without evidence, to be open to new knowledge, ideas, ways.
The perception of information occurs in three stages, which corresponds to the following stages of the lesson:
- preparatory - call stage;
- perception of the new - the semantic stage (or the stage of the realization of the meaning);
- appropriation of information is the stage of reflection.
Students work in groups, compare their assumptions with information obtained in the course of working with the textbook, plotting functions and descriptions of their properties, make changes to the proposed “Do you believe that ...” table, share thoughts with the class, discuss the answers to each question . At the call stage, it is clarified in what cases, in the performance of which tasks, the properties of the logarithmic function can be applied. At the stage of understanding the content, work is underway to recognize graphs of logarithmic functions, find the domain of definition, and determine the monotonicity of functions.
To expand knowledge on the subject under study, students are offered the text "Application of the logarithmic function in nature and technology." We use to maintain interest in the topic. Pupils work in groups, making clusters "Application of the logarithmic function". Then the clusters are defended and discussed.
Sinkwine is used as a creative form of reflection, which develops the ability to summarize information, express complex ideas, feelings and ideas in a few words.
Equipment: PowerPoint presentation, interactive whiteboard, handouts (cards, text material, tables), sheets of paper in a cage.
During the classes
Call stage:
Teacher introduction. We are working on mastering the topic "Logarithms". What do we currently know and can do?
Student responses.
We know Keywords: definition, properties of the logarithm, basic logarithmic identity, formulas for the transition to a new base, areas of application of logarithms.
We know how: calculate logarithms, solve the simplest logarithmic equations, perform transformations of logarithms.
What concept is closely related to the concept of the logarithm? (with the concept of degree, since the logarithm is an exponent)
Assignment to students. Using the concept of the logarithm, fill in any two tables with a > 1 and at 0 < a< 1 (Appendix No. 1)
Checking the work of groups.
What are the expressions shown? (exponential equations, exponential functions)
Assignment to students. Solve exponential equations using variable expression X through a variable at.
As a result of this work, the following formulas are obtained:
In the resulting expressions, we swap X And at. What happened to us?
How would you call these functions? (logarithmic, since the variable is under the sign of the logarithm). How to write this function in general form?
The topic of our lesson is “Logarithmic function, its properties and graph”.
A logarithmic function is a function of the form , where A- a given number, a>0, a≠1.
Our task is to learn how to build and explore graphs of logarithmic functions, apply their properties.
There are question cards on the tables. They all begin with the words "Do you believe that ..."
The answer to the question can only be "yes" or "no". If “yes”, then to the right of the question in the first column put a “+” sign, if “no”, then a “-” sign. If in doubt, put a sign "?".
Work in pairs. Working time 3 minutes. (Appendix No. 2)
After listening to the students' answers, the first column of the pivot table on the board is filled in.
Content comprehension stage(10 min).
Summing up the work with the questions of the table, the teacher prepares the students for the idea that when answering questions, we do not yet know whether we are right or not.
Task for groups. Answers to questions can be found by studying the text of §4 pp.240-242. But I suggest not just reading the text, but choosing one of the four previously obtained functions: plotting its graph and identifying the properties of the logarithmic function from the graph. Each member of the group does this in a notebook. And then, on a large sheet in a cell, a graph of the function is built. After the work is completed, a representative of each group will defend their work.
Assignment to groups. Generalize function properties for a > 1 And 0 < a< 1 (Appendix No. 3)
Axis OU is the vertical asymptote of the graph of the logarithmic function and in the case when a>1, and in the case when 0
Function Graph passes through a point with coordinates (1;0)
Assignment to groups. Prove that exponential and logarithmic functions are mutually inverse.
Students in the same coordinate system depict a graph of a logarithmic and exponential function
Consider two functions simultaneously: the exponential y = a x and logarithmic y = log a x.
Figure 2 schematically shows the graphs of functions y = a x And y = log a x in case when a>1.
Figure 3 schematically shows the graphs of functions y = a x And y = log a x in case when 0 < a < 1.
The following assertions are true.
- Function Graph y = log a x symmetric to the graph of the function y \u003d ax with respect to the straight line y = x.
- The set of function values y = a x is the set y>0, and the domain of the function y = log a x is the set x>0.
- Axis Oh is the horizontal asymptote of the graph of the function y = a x, and the axis OU is the vertical asymptote of the graph of the function y = log a x.
- Function y = a x increases with a>1 and function y = log a x also increases with a>1. Function y = a x decreases at 0<а<1 and function y = log a x also decreases with 0<а<1
Therefore, indicative y = a x and logarithmic y = log a x functions are mutually inverse.
Function Graph y = log a x called the logarithmic curve, although in fact a new name could not be invented. After all, this is the same exponent that serves as a graph of the exponential function, only differently located on the coordinate plane.
Reflection stage. Preliminary summing up.
Let's go back to the questions discussed at the beginning of the lesson and discuss the results.. Let's see, maybe our opinion after work has changed.
Students in groups compare their assumptions with information obtained in the course of working with the textbook, plotting functions and descriptions of their properties, make changes to the table, share thoughts with the class, and discuss the answers to each question.
Call stage.
What do you think, in what cases, when performing what tasks, can the properties of the logarithmic function be applied?
Intended student responses: solving logarithmic equations, inequalities, comparing numerical expressions containing logarithms, constructing, transforming, and exploring more complex logarithmic functions.
Content comprehension stage.
Job on recognizing graphs of logarithmic functions, finding the domain of definition, determining the monotonicity of functions. (Appendix No. 4)
Answers.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 1)a, 2)b, 3)c | 1) a, 2) c, 3) a | a, in | V | B, C | A)< б) > | A)<0 б) <0 |
To expand knowledge on the subject under study, students are offered the text "Application of the logarithmic function in nature and technology." (Appendix No. 5) We use technological method "Cluster" to maintain interest in the topic.
“Does this function find application in the world around us?”, we will answer this question after working on the text about the logarithmic spiral.
Compilation of the cluster "Application of the logarithmic function". Students work in groups, forming clusters. Then the clusters are defended and discussed.
Cluster example.
Reflection
- What did you have no idea about until today's lesson, and what is now clear to you?
- What have you learned about the logarithmic function and its applications?
- What difficulties did you encounter while completing the assignments?
- Highlight the question that is less clear to you.
- What information are you interested in?
- Compose the syncwine "logarithmic function"
- Evaluate the work of your group (Appendix No. 6 "Group performance evaluation sheet")
Sincwine.
- logarithmic function
- Unlimited, monotonous
- Explore, compare, solve inequalities
- Properties depend on the value of the base of the logarithmic function
- Exhibitor
Homework:§ 4 pp. 240-243, no. 69-75 (even)
Literature:
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