Give the definition of a logarithmic function. “Logarithmic function, its properties and graph

The logarithmic function is based on the concept of logarithm and the properties of the exponential function, where (the base of the power a is greater than zero and not equal to one).

Definition:

The logarithm of b to base a is the exponent to which base a must be raised to get b.

Examples:

Let us remind you basic rule: to get the number under the logarithm, you need to raise the base of the logarithm to a power - the value of the logarithm:

Let us recall the important features and properties of the exponential function.

Let's consider the first case, when the base of the degree is greater than one: :

Rice. 1. Graph of an exponential function, the base of the power is greater than one

Such a function increases monotonically throughout its entire domain of definition.

Consider the second case, when the base of the degree is less than one:

Rice. 2. Graph of an exponential function, base of exponent less than one

Such a function decreases monotonically throughout its entire domain of definition.

In any case, the exponential function is monotonic, takes all positive values and, due to its monotonicity, reaches each positive value for a single value of the argument. That is, the function reaches each specific value with a single value of the argument, the root of the equation is the logarithm:

Essentially, we got the inverse function. A direct function is when we have an independent variable x (argument), dependent variable y (function), we set the value of the argument and from it we get the value of the function. Inverse function: let the independent variable be y, because we have already stated that each positive value of y corresponds to a single value x, the definition of the function is respected. Then x becomes the dependent variable.

For a monotonic direct function there is an inverse function. The essence of the functional dependence will not change if we introduce a redesignation:

We get:

But we are more accustomed to denoting the independent variable by x, and the dependent variable by y:

Thus, we have obtained a logarithmic function.

We use the general rule for obtaining the inverse function for a specific exponential function.

The given function increases monotonically (according to the properties of the exponential function), which means that there is a function inverse to it. We remind you that to obtain it you need to perform two steps:

Express x in terms of y:

Swap x and y:

So, we got the function inverse to the given one: . As is known, the graphs of the direct and inverse functions are symmetrical with respect to the straight line y=x. Let's illustrate:

Rice. 3. Graphs of functions and

This problem can be solved in a similar way and is valid for any degree base.

Let's solve the problem when

The given function decreases monotonically, which means that there is an inverse function. Let's get it:

Express x in terms of y:

Swap x and y:

So, we got the function inverse to the given one: . As is known, the graphs of the direct and inverse functions are symmetrical with respect to the straight line y=x. Let's illustrate:

Rice. 4. Graphs of functions and

Note that we obtained logarithmic functions as the inverse of exponential functions.

The forward and inverse functions have much in common, but there are also differences. Let's look at this in more detail using the functions and as an example.

Rice. 5. Graphs of functions (left) and (right)

Properties of the direct (exponential) function:

Domain: ;

Range of values: ;

The function increases;

Convex downwards.

Properties of the inverse (logarithmic) function:

Domain: ;

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 TransStroyTech"

Ministry of Education of Chuvashia)

Methodological development

ODP. 01 Mathematics

"Logarithmic function. Properties and schedule"

Cheboksary - 2016

Explanatory note……………….................................................... ......…………………………………….….…3

Theoretical justification and methodological implementation…………….….................................4-10

Conclusion…………………………………………………………….......................... .........................………....eleven

Applications……………………………………………………………………………….......................... ..........................………...13

Explanatory note

Methodological development of a lesson module in the discipline “Mathematics” on the topic “Logarithmic function. Properties and graph" from the section "Roots, powers 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 content and main provisions.

The purpose of studying this topic is to learn the concept of a logarithmic function, study its basic properties, learn to build a graph of a logarithmic function and learn to see a logarithmic spiral in the world around us.

The program material for this lesson is based on knowledge of mathematics. The methodological development of the lesson module was compiled for conducting theoretical classes on the topic: “Logarithmic function. Properties and schedule" -1 hour. During the practical lesson, students consolidate their acquired knowledge: definitions of functions, their properties and graphs, transformations of graphs, continuous and periodic functions, inverse functions and their graphs, logarithmic functions.

The methodological development is intended to provide methodological assistance to students when studying the lesson module on the topic “Logarithmic function. Properties and schedule". As extracurricular independent work, students can prepare, with the help of additional sources, a message on the topic “Logarithms and their application in nature and technology,” crosswords and puzzles. The educational knowledge and professional competencies acquired while studying the topic “Logarithmic functions, their properties and graphs” will be applied when studying the following sections: “Equations and Inequalities” and “Principles of Mathematical Analysis”.

Didactic structure of the lesson:

Subject:« Logarithmic function. Properties and graph »

Type of activity: Combined.

Lesson objectives:

Educational- formation of knowledge in mastering the concept of a logarithmic function, properties of a logarithmic function; use graphs to solve problems.

Developmental- development of mental operations through concretization, development of visual memory, the need for self-education, to promote the development of cognitive processes.

Educational- fostering cognitive activity, a 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 by Sh.A Alimov “Algebra and the beginnings of analysis” grades 10-11. Publishing house "Prosveshcheniye".

Intrasubject connections: exponential function and logarithmic function.

Interdisciplinary connections: algebra and mathematical analysis.

Studentmust know:

definition of 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 logarithms;

determine the position of a point on the graph by its coordinates and vice versa;

apply the properties of a logarithmic function when constructing graphs;

Perform graph transformations.

Lesson plan

1. Organizational moment (1 min).

2. Setting the goals and objectives of the lesson. Motivation for students' learning activities (1 min).

3. Stage of updating basic knowledge and skills (3 min).

4. Checking homework (2 min).

5. Stage of assimilation of new knowledge (10 min).

6. Stage of consolidating new knowledge (15 min).

7. Monitoring the material learned in the lesson (10 min).

8. Summing up (2 min).

9. Stage of informing students about homework (1 min).

During the classes:

1. Organizational moment.

Includes the teacher greeting the class, preparing the room for the lesson, and checking on absentees.

2. Setting goals and objectives for the lesson.

Today we will talk about the concept of a logarithmic function, draw a graph of the 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? Draw schematically on the board.

Give the definition of an exponential function.

What is the root of an exponential equation?

Define logarithm?

What are the properties of logarithms?

What is the main logarithmic identity?

4. Checking homework.

Students open their notebooks and show the solved exercises. Ask questions that arose while doing homework.

5. Stage of assimilation of new knowledge.

Teacher: Open your 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, .

Let's look at constructing a graph of this function using a specific example.

Let's build graphs of functions and .

Note 1: The logarithmic function is the inverse of the exponential function, where . Therefore, their graphs are symmetrical relative to the bisector of coordinate angles I and III (Fig. 1).

Based on the definition of the logarithm and the type of graphs, we will identify the properties of the logarithmic function:

1) Scope of definition: , because by definition of the logarithm x>0.

2) Function range: .

3) The logarithm of one is equal to zero, the logarithm of the base is equal to one: , .

4) Function , increases in the interval (Fig. 1).

5) Function , decrease in the interval (Fig. 1).

6) Intervals of constancy of signs:

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) (§18 Alimov Sh.A. “Algebra and the beginnings of analysis” 10-11 grade).

1) because the function increases.

3), because the function decreases.

1) , because and .

3) , because and .

1) , because , , then .

3) , because 10> 1, then .

1) decreases

3)increases.

7. Summing up.

- Today we did a good job in class! What new did you learn in class today?

(New type of function - logarithmic function)

State the definition of a logarithmic function.

(The function y = logax, (a > 0, a ≠ 1) is called a logarithmic function)

Well done! Right! Name the properties of the logarithmic function.

(domain of definition of a function, set of function values, monotonicity, constancy of sign)

8. Control of the material learned in the lesson.

Teacher: Let's find out how well you have mastered the topic “Logarithmic function. Properties and schedule". 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 are given 10 minutes to complete the test.

9. The stage of informing students about homework.

Writing on the board and in diaries: Alimov Sh.A. “Algebra and beginnings of analysis” grades 10-11. §18 No. 318 - No. 322 (even)

Conclusion

In the course of using the methodological development, we achieved all our goals and objectives. In this methodological development, all the properties of the logarithmic function were considered, thanks to which students learned to transform expressions containing logarithms and build graphs of logarithmic functions. Completing practical tasks helps to consolidate the studied material, and monitoring the testing of 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 educational information on the topic, generalize and systematize knowledge, apply the properties of logarithms and logarithmic functions when solving various logarithmic equations and inequalities.

Alimov Sh. A., Kolyagin Yu. M., Sidorov Yu. V., Fedorova N. E., Shabunin M. I. under the scientific guidance of Academician Tikhonov A. N. Algebra and the beginnings of mathematical analysis 10 - 11 grades. - 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 grades - M., 2006.

Kolyagin Yu.M., Tkacheva M.V., Federova N.E. and others, ed. Zhizhchenko A.B. Algebra and beginnings of mathematical analysis (basic and specialized levels). 10 grades - 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 s.

Internet resources:

http://school-collection.edu.ru - Electronic textbook “Mathematics in

school, XXI 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:

A mark of “3” (satisfactory) is given for any 2 correctly completed examples.

The mark “4” (good) is given if any 3 examples are completed correctly.

The mark “5” (excellent) is given for all 4 correctly completed examples.

The section on logarithms is of great importance in the school course “Mathematical Analysis”. Problems for logarithmic functions are based on different principles than problems for inequalities and equations. Knowledge of the definitions and basic properties of the concepts of logarithm and logarithmic function will ensure the successful solution of typical USE problems.

Before we begin to explain what a logarithmic function is, it is worth looking at the definition of a logarithm.

Let's look at a specific example: a log a x = x, where a › 0, a ≠ 1.

The main properties of logarithms can be listed in several points:

Logarithm

Logarithmation is a mathematical operation that allows, using the properties of a concept, to find the logarithm of a number or expression.

Examples:

Logarithm function and its properties

The logarithmic function has the form

Let us immediately note that the graph of a function can be increasing when a › 1 and decreasing when 0 ‹ a ‹ 1. Depending on this, the function curve will have one form or another.

Here are the properties and method of plotting logarithms:

the domain of f(x) is the set of all positive numbers, i.e. x can take any value from the interval (0; + ∞);
ODZ function is the set of all real numbers, i.e. y can be equal to any number from the interval (— ∞; +∞);
if the base of the logarithm a › 1, then f(x) increases throughout the entire domain of definition;
if the base of the logarithm is 0 ‹ a ‹ 1, then F is decreasing;
the logarithmic function is neither even nor odd;
the graph curve always passes through the point with coordinates (1;0).

It’s very easy to build both types of graphs; let’s look at the process using an example

First you need to remember the properties of the simple logarithm and its functions. With their help, you need to build a table for specific values of x and y. Then you should mark the resulting points on the coordinate axis and connect them with a smooth line. This curve will be the required graph.

The logarithmic function is the inverse of the exponential function given by y= a x. To verify this, it is enough to draw both curves on the same coordinate axis.

It is obvious that both lines are mirror images of each other. By constructing the straight line y = x, you can see the axis of symmetry.

In order to quickly find the answer to the problem, you need to calculate the values of the points for y = log 2⁡ x, and then simply move the origin of the coordinate point three divisions down along the OY axis and 2 divisions to the left along the OX axis.

As proof, let's build a calculation table for the points of the graph y = log 2 ⁡(x+2)-3 and compare the obtained values with the figure.

As you can see, the coordinates from the table and the points on the graph coincide, therefore, the transfer along the axes was carried out correctly.

Examples of solving typical Unified State Exam problems

Most of the test problems can be divided into two parts: searching for the domain of definition, indicating the type of function based on the graph drawing, determining whether the function is increasing/decreasing.

To quickly answer tasks, it is necessary to clearly understand that f(x) increases if the logarithm exponent a › 1, and decreases if 0 ‹ a ‹ 1. However, not only the base, but also the argument can greatly influence the shape of the function curve.

F(x) marked with a checkmark are correct answers. Examples 2 and 3 raise doubts in this case. The “-” sign in front of log changes increasing to decreasing and vice versa.

Therefore, the graph y=-log 3⁡ x decreases over the entire domain of definition, and y= -log (1/3) ⁡x increases, despite the fact that the base 0 ‹ a ‹ 1.

Answer: 3,4,5.

Answer: 4.

These types of tasks are considered easy and are scored 1-2 points.

Task 3.

Determine whether the function is decreasing or increasing and indicate the domain of its definition.

Y = log 0.7 ⁡(0.1x-5)

Since the base of the logarithm is less than one but greater than zero, the function of x is decreasing. According to the properties of the logarithm, the argument must also be greater than zero. Let's solve the inequality:

Answer: domain of definition D(x) – interval (50; + ∞).

Answer: 3, 1, OX axis, right.

Such tasks are classified as average and are scored 3 - 4 points.

Task 5. Find the range of values for a function:

From the properties of the logarithm it is known that the argument can only be positive. Therefore, we will calculate the range of acceptable values of the function. To do this, you will need to solve a system of two inequalities.

The basic properties of the logarithm, logarithm graph, domain of definition, set of values, basic formulas, increasing and decreasing are given. Finding the derivative of a logarithm is considered. As well as integral, power series expansion and representation using complex numbers.

Definition of logarithm

Logarithm with base a is a function of y (x) = log a x, inverse to the exponential function with base a: x (y) = a y.

Decimal logarithm is the logarithm to the base of a number 10 : log x ≡ log 10 x.

Natural logarithm is the logarithm to the base of e: ln x ≡ log e x.

2,718281828459045... ;
.

The graph of the logarithm is obtained from the graph of the exponential function by mirroring it with respect to the straight line y = x. On the left are graphs of the function y (x) = log a x for four values logarithm bases: a = 2 , a = 8 , a = 1/2 and a = 1/8 . The graph shows that when a > 1 the logarithm increases monotonically. As x increases, growth slows down significantly. At 0 < a < 1 the logarithm decreases monotonically.

Properties of the logarithm

Domain, set of values, increasing, decreasing

The logarithm is a monotonic function, so it has no extrema. The main properties of the logarithm are presented in the table.


Domain	0 < x < + ∞	0 < x < + ∞
Range of values	- ∞ < y < + ∞	- ∞ < y < + ∞
Monotone	monotonically increases	monotonically decreases
Zeros, y = 0	x = 1	x = 1
Intercept points with the ordinate axis, x = 0	No	No
	+ ∞	- ∞
	- ∞	+ ∞

Private values

The logarithm to base 10 is called decimal logarithm and is denoted as follows:

Logarithm to base e called natural logarithm:

Basic formulas for logarithms

Properties of the logarithm arising from the definition of the inverse function:

The main property of logarithms and its consequences

Base replacement formula

Logarithm is the mathematical operation of taking a logarithm. When taking logarithms, products of factors are converted into sums of terms.

Potentiation is the inverse mathematical operation of logarithm. During potentiation, a given base is raised to the degree of expression over which potentiation is performed. In this case, the sums of terms are transformed into products of factors.

Proof of basic formulas for logarithms

Formulas related to logarithms follow from formulas for exponential functions and from the definition of an inverse function.

Consider the property of the exponential function
.
Then
.
Let's apply the property of the exponential function
:
.

Let us prove the base replacement formula.
;
.
Assuming c = b, we have:

Inverse function

The inverse of a logarithm to base a is an exponential function with exponent a.

If , then

Derivative of logarithm

Derivative of the logarithm of modulus x:
.
Derivative of nth order:
.
Deriving formulas > > >

To find the derivative of a logarithm, it must be reduced to the base e.
;
.

Integral

The integral of the logarithm is calculated by integrating by parts: .
So,

Expressions using complex numbers

Consider the complex number function z:
.
Let's express a complex number z via module r and argument φ :
.
Then, using the properties of the logarithm, we have:
.
Or

However, the argument φ not uniquely defined. If you put
, where n is an integer,
then it will be the same number for different n.

Therefore, the logarithm, as a function of a complex variable, is not a single-valued function.

Power series expansion

When the expansion takes place:

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.

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 types of logarithms are:

If we consider the 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

When the equality is true

(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. Logarithmic scale

Logarithms to base 10 (symbol: lg a) before the invention of calculators were widely used for calculations. The uneven scale of decimal logarithms is usually marked on slide rules as well. A similar scale is widely used in various fields of science, for example:

Chemistry - activity of hydrogen ions ().
Music theory - a scale of notes, in relation to the frequencies of musical notes.

The logarithmic scale is also widely used to identify the exponent in power relations and the coefficient in the exponent. In this case, a graph constructed 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

A 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 first order) is obtained at z= 1, singular 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 sketch

Real logarithm

The need for complex calculations grew rapidly in the 16th century, and much of the difficulty involved multiplying and dividing multi-digit numbers. At the end of the century, several mathematicians, almost simultaneously, came up with the idea: to replace labor-intensive multiplication with simple addition, using special tables to compare the geometric and arithmetic progressions, with the geometric one being the original one. Then division is automatically replaced by the 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 engineer's tool.

A close to modern understanding of logarithmation - as the inverse operation of raising to a power - first appeared with Wallis and Johann Bernoulli, and was finally legitimized 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 is also credited with 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 because the very concept of a logarithm was not yet clearly defined. The discussion on this issue took place first between Leibniz and Bernoulli, and in the middle of the 18th century - between d'Alembert and Euler. Bernoulli and d'Alembert believed that it should be determined log(-x) = log(x). The complete theory of 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

From the properties of the logarithm it follows that instead of labor-intensive multiplication of multi-digit numbers, it is enough to find (from tables) and add their logarithms, and then, using the same tables, perform potentiation, that is, find the value of the result from its logarithm. Doing division differs only in that logarithms are subtracted. Laplace said that the invention of logarithms “extended the life of astronomers” by greatly speeding up the process of calculations.

When moving the decimal point in a number to n digits, the value of the decimal logarithm of this number changes to n. For example, log8314.63 = log8.31463 + 3. It follows that it is enough to compile 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 logarithms of trigonometric functions, and with errors. Independently of him, Joost Bürgi, a friend of Kepler (), published his tables. In 1617, Oxford mathematics professor Henry Briggs published tables that already included decimal logarithms of the numbers themselves, from 1 to 1000, with 8 (later 14) digits. But there were also errors in Briggs' tables. The first error-free edition based on the Vega tables () appeared only in 1857 in Berlin (Bremiwer tables).

In Russia, the first tables of logarithms were published in 1703 with the participation of L. F. Magnitsky. Several collections of logarithm tables were published in the USSR.

Bradis V. M. Four-digit math tables. 44th edition, M., 1973.