General view of the logarithmic function. logarithmic function

"Logarithmic function, its properties and graph".

Byvalina L.L., teacher of mathematics, MBOU secondary school, Kiselevka village, Ulchsky district, Khabarovsk Territory

Algebra grade 10

Lesson topic: "Logarithmic function, its properties and graph."

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

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 (Appendix No. 1)

X				1	2	4	8	16		X				1	2	4	8	16
	-3	-2	-1	0	1	2	3	4			3	2	1	0	-1	-2	-3	-4

X			1	3	9		X			1	3	9
	-2	-1	0	1	2			2	1	0	-1	-2

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)

№ p/p	Questions:	BUT	B	AT
Do you believe that...
1.	The y-axis is the vertical asymptote of the graph of the logarithmic function.			+
2.	exponential and logarithmic functions mutually inverse functions			+
3.	Graphs of the exponential y \u003d a x and the logarithmic functions are symmetrical with respect to the straight line y \u003d x.			+
4.	The domain of the logarithmic function is the entire number line X (-∞, +∞)			-
5.	The range of the logarithmic function is the interval at (0, +∞)			-
6.	The monotonicity of the logarithmic function depends on the base of the logarithm			+
7.	Not every graph of a logarithmic function passes through a point with coordinates (1; 0).			-
8.	The logarithmic curve is the same exponential, only differently located in the coordinate plane.			+
9.	The convexity of a logarithmic function does not depend on the base of the logarithm.			-
10.	The logarithmic function is neither even nor odd.			+
11.	The logarithmic function has the largest value and does not have the smallest value when a > 1 and vice versa when 0 a			-

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 propose not just to read the text, but to choose one of the four previously obtained functions: ,, , , build its graph and identify 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 (Appendix No. 3)

Function Properties y = log a x at a > 1.

Function Properties y = log a x , at 0 .

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 y = log a x 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
fig.3.

The following assertions are true.

Function Graph y = log a X symmetrical to the graph of the function y \u003d a x 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 0y = log a X also decreases with 0

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.
Work on recognizing graphs of logarithmic functions, finding the domain of definition, determining the monotonicity of functions. (Appendix No. 4)

1. Find the scope of the function:

1)at= log 0,3 X 2) at= log 2 (x-1) 3) at= log 3 (3-x)

(0; +∞) b) (1;+∞) c) (-∞; 3) d) (0;1]

2. At what values X makes sense function: 1) at= log 3 X 2 2)at= log 5 (-X) 3)at= lg │x│

a) x≠0 b) x>0 in) .

1	2	3	4	5	6	7
1)a, 2)b, 3)c	1) a, 2) c, 3) a	a, in	in	B, C	a)	a)

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.

Application of the logarithmic function

Nature

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.

Homework:§ 4 pp. 240-243, no. 69-75 (even)

Literature:

Azevich A.I. Twenty Lessons of Harmony: Humanities and Mathematics Course. - M.: School-Press, 1998.-160 p.: ill. (Library of the journal "Mathematics at School". Issue 7.)
Zair.Bek S.I. The development of critical thinking in the classroom: a guide for general education teachers. institutions. - M. Education, 2011. - 223 p.
Kolyagin Yu.M. Algebra and the beginnings of analysis. Grade 10: textbook. for general education institutions: basic and profile levels. – M.: Enlightenment, 2010.
Korchagin V.V. USE-2009. Maths. Thematic training tasks. – M.: Eksmo, 2009.
USE-2008. Maths. Thematic training tasks / Koreshkova T.A. and others. - M .: Eksmo, 2008

The section of logarithms is of great importance in the school course "Mathematical Analysis". Tasks for logarithmic functions are based on other principles than tasks 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 proceeding to explain what a logarithmic function is, it is worth referring to 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

Logarithm 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

We note right away that the graph of a function can be increasing for a › 1 and decreasing for 0 ‹ a ‹ 1. Depending on this, the function curve will have one form or another.

Here are the properties and method for plotting graphs of 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 functions - 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 over 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).

Building both types of graphs is very simple, let's look at the process using an example

First you need to remember the properties of a simple logarithm and its function. With their help, you need to build a table for specific x and y values. Then, on the coordinate axis, the obtained points should be marked and connected by 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.

Obviously, both lines are mirror images of each other. By constructing a 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 points three divisions down the OY axis and 2 divisions to the left along the OX axis.

As proof, we will 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 match, therefore, the transfer along the axes was carried out correctly.

Examples of solving typical USE problems

Most of the test tasks can be divided into two parts: finding the domain of definition, specifying the type of function according to the graph drawing, determining whether the function is increasing / decreasing.

For a quick answer to tasks, it is necessary to clearly understand that f (x) increases if the exponent of the logarithm a > 1, and decreases - when 0 ‹ a ‹ 1. However, not only the base, but also the argument can greatly affect the form of the function curve.

F(x) marked with a check mark are the correct answers. Doubts in this case are caused by examples 2 and 3. 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 is 0 ‹ a ‹ 1.

Answer: 3,4,5.

Answer: 4.

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

Task 3.

Determine whether the function is decreasing or increasing and indicate the scope 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: the domain of definition D(x) is the interval (50; + ∞).

Answer: 3, 1, OX axis, to the right.

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

Task 5. Find the range for a function:

It is known from the properties of the logarithm that the argument can only be positive. Therefore, we calculate the area of admissible values of the function. To do this, it will be necessary to solve a system of two inequalities.

The main properties of the logarithm, the graph of the logarithm, the domain of definition, the set of values, the basic formulas, the increase and decrease are given. Finding the derivative of the logarithm is considered. As well as integral, power series expansion and representation by means of complex numbers.

Definition of logarithm

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

Decimal logarithm is the logarithm to the base of the 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 mirror reflection about the straight line y \u003d x. On the left are graphs of the function y (x) = log x for four values bases of the logarithm:a= 2 , a = 8 , a = 1/2 and a = 1/8 . The graph shows that for a > 1 the logarithm is monotonically increasing. As x increases, the growth slows down significantly. At 0 < a < 1 the logarithm is monotonically decreasing.

Properties of the logarithm

Domain, set of values, ascending, descending

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


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

Private values

The base 10 logarithm is called decimal logarithm and is marked like this:

base logarithm e called natural logarithm:

Basic logarithm formulas

Properties of the logarithm following 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 the logarithm. When taking a logarithm, the products of factors are converted to sums of terms.

Potentiation is the mathematical operation inverse to logarithm. When potentiating, the given base is raised to the power of the expression on which the potentiation is performed. In this case, the sums of terms are converted into products of factors.

Proof of the 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
.
Apply the property of the exponential function
:
.

Let us prove the base change formula.
;
.
Setting c = b , we have:

Inverse function

The reciprocal of the base a logarithm is the exponential function with exponent a.

If , then

Derivative of the logarithm

Derivative of logarithm modulo x :
.
Derivative of the nth order:
.
Derivation of 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 in terms of 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 clearly defined. If we 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

For , the expansion takes place:

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of Mathematics for Engineers and Students of Higher Educational Institutions, Lan, 2009.

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:

Topic:« 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 a , 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.

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 concept of the logarithm itself 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

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 professor of mathematics 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.

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