Logarithmic function. Big encyclopedia of oil and gas

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 describing 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)

Answers.

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.

1. logarithmic function
2. Unlimited, monotonous
3. Explore, compare, solve inequalities
4. Properties depend on the value of the base of the logarithmic function
5. Exhibitor

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

Literature:

1. 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.)
2. Zair-Bek S.I. The development of critical thinking in the classroom: a guide for general education teachers. institutions. - M. Education, 2011. - 223 p.
3. Kolyagin Yu.M. Algebra and the beginnings of analysis. Grade 10: textbook. for general education institutions: basic and specialized levels. – M.: Enlightenment, 2010.
4. Korchagin V.V. USE-2009. Maths. Thematic training tasks. – M.: Eksmo, 2009.
5. USE-2008. Maths. Thematic training tasks / Koreshkova T.A. and others. - M .: Eksmo, 2008.

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

Definition:

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

Examples:

Recall ground 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:

Recall important features and properties of an exponential function.

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 degree is greater than one

Such a function is monotonically increasing over 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, the base of the degree is less than one

Such a function is monotonically decreasing over 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 with a single value of the argument. That is, each specific value the function reaches with a single value of the argument , the root of the equation is the logarithm:

In fact, we got the inverse function. A direct function is when we have an independent variable x (argument), a dependent variable y (function), we set the value of the argument and use it to get the value of the function. Inverse function: let y be the independent variable, because we have already stipulated that each positive value of y corresponds to a single value of 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 functional dependency will not change if we introduce a renaming:

We get:

But we are more accustomed to denote the independent variable as x, and the dependent variable as y:

Thus, we have obtained a logarithmic function.

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

The given function is monotonically increasing (according to the properties of the exponential function), which means that there is a function inverse to it. We remind you that in order to receive it, you must perform two steps:

Express x in terms of y:

Swap x and y:

So, we got the function inverse to the given one: . As you know, the graphs of the direct and inverse functions are symmetrical with respect to the straight line y \u003d x. let's illustrate:

Rice. 3. Graphs of functions and

This problem is solved in a similar way and is valid for any base of the degree.

Let's solve the problem with

The given function is monotonically decreasing, which means that there is an inverse function to it. 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 you know, the graphs of the direct and inverse functions are symmetrical with respect to the straight line y \u003d x. let's illustrate:

Rice. 4. Graphs of functions and

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

Direct and inverse functions have much in common, but there are also differences. Let's consider this in more detail using the example of functions and .

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

Properties of a direct (exponential) function:

Domain: ;

Range of values: ;

The function is increasing;

Curved down.

Properties of the inverse (logarithmic) function:

Domain: ;

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

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

Logarithmic tables

It follows from the properties of the logarithm that instead of the time-consuming multiplication of multi-valued 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.

"Logarithmic function, its properties and graph".

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)

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)

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 but- 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 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 describing 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)

1. 
   (0; +∞) b) (1;+∞) c) (-∞; 3) d) (0;1]

but) x≠0 b) x>0 in) .

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

Literature:

1. 
   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.)
2. 
   Zair.Bek S.I. The development of critical thinking in the classroom: a guide for general education teachers. institutions. - M. Education, 2011. - 223 p.
3. 
   Kolyagin Yu.M. Algebra and the beginnings of analysis. Grade 10: textbook. for general education institutions: basic and specialized levels. – M.: Enlightenment, 2010.
4. 
   Korchagin V.V. USE-2009. Maths. Thematic training tasks. – M.: Eksmo, 2009.
5. 
   USE-2008. Maths. Thematic training tasks / Koreshkova T.A. and others. - M .: Eksmo, 2008