Logarithmic function, its graph and properties. Function y=log a x, its properties and graph (continued)

Graph of a function passes through a point with coordinates (1;0)

Group assignment. Prove that the exponential and logarithmic functions are mutually inverse.

Students draw a graph of a logarithmic and exponential function in the same coordinate system

Let us consider two functions simultaneously: exponential y = a x and logarithmic y = log a x.

Figure 2 schematically shows the graphs of the functions y = a x And y = log a x in case a>1.

Figure 3 schematically shows the graphs of the functions y = a x And y = log a x in case 0 < a < 1.

The following statements are true.

• Graph of a function y = log a x is symmetrical to the graph of the function y = ax relative to the straight line y = x.
• Function value set y = a x is a set y>0, and the domain of definition of the function y = log a x is a 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 at 0<а<1

Therefore, indicative y = a x and logarithmic y = log a x the functions are mutually inverse.

Graph of a function y = log a x called a logarithmic curve, although in fact a new name could not be invented. After all, this is the same exponent that serves as the graph of the exponential function, only located differently on the coordinate plane.

Reflection stage. Preliminary summary.

Let's return to the questions discussed at the beginning of the lesson and discuss the results obtained. Let's see, maybe our opinion has changed after work.

Students in groups compare their assumptions with the information obtained from working with the textbook, constructing graphs of functions and descriptions of their properties, make changes to the table, share their thoughts with the class, and discuss the answers to each question.

Call stage.

In what cases do you think, when performing what tasks can the properties of a logarithmic function be applied?

Expected 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.

To expand knowledge on the issue 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.

“Is this function applicable in the world around us?”, we will answer this question after working on the text about the logarithmic spiral.

Compiling the cluster “Application of the logarithmic function.” Students work in groups, making clusters. Then the clusters are protected and discussed.

Cluster example.

Reflection

• What did you have no idea about before today's lesson, and what has now become clear to you?
• What have you learned about the logarithmic function and its applications?
• What difficulties did you encounter while completing the tasks?
• Highlight the question that was less clear to you.
• What information interested you?
• Compose a logarithmic function syncwine
• Evaluate your group's work (Appendix No. 6 “Group Performance Evaluation Sheet”)

Sinkwine.

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 p.240-243, No. 69-75 (even)

Literature:

1. Azevich A.I. Twenty lessons of harmony: Humanities and mathematics course. - M.: Shkola-Press, 1998.-160 p.: ill. (Library of the journal “Mathematics at School”. Issue 7.)
2. Zair-Bek S.I. Development of critical thinking in the classroom: a manual for general education teachers. institutions. – M. Education, 2011. – 223 p.
3. Kolyagin Yu.M. Algebra and the beginnings of analysis. 10th grade: textbook. for general education institutions: basic and profile levels. – M.: Education, 2010.
4. Korchagin V.V. Unified State Exam 2009. Mathematics. Thematic training tasks. – M.: Eksmo, 2009.
5. Unified State Exam 2008. Mathematics. Thematic training tasks/ Koreshkova T.A. and others. - M.: Eksmo, 2008.

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. Study the basic properties of the logarithmic function. Develop the ability to plot a logarithmic function.

Developmental: Develop the ability to highlight the main thing, compare, generalize. To form a graphic culture among students.

Educational: Show the relationship between mathematics and the surrounding reality. Develop communication skills, dialogue, and the ability to work in a team.

Lesson type: Combined

Teaching methods: Partially search, interactive.

During the classes.

1.Updating past experience:

Students are offered oral exercises using the definition of the logarithm, its properties, formulas for moving to a new base, solving the simplest logarithmic and exponential equations, examples of finding the range of acceptable values ​​for logarithmic expressions

Oral exercisesOral work.

1) Calculate using the definition of logarithm: log 2 8; log 4 16;.

2) Calculate using the basic logarithmic identity:

3) Solve the equation using the definition:

4) Find out at what values ​​of x the expression makes sense:

5) Find the value of the expression using the properties of logarithms:

2. Study the topic. Students are asked to solve exponential equations: 2 x =y; () x = y. by expressing the variable x in terms of the variable y. As a result of this work, formulas are obtained that define functions unfamiliar to students. ,. Question : “What would you call this function?” students say that it is logarithmic, since the variable is under the logarithm sign: .

Question . Define a function. Definition: A function given by the formula y=log a x is called logarithmic with base a (a>0, a 1)

III. Function study y=log a x

More recently, we introduced the concept of the logarithm of a positive number to a positive and non-1 base a. For any positive number, you can find the logarithm to a given base. But then you should think about a function of the form 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, a 1)

Basic 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 calledR+. An obvious property, since every positive number has a logarithm to base a.D(f)=R+

2. The range 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 ageno at a>1, and decreases at 0<х<1.

5 . The function is not even or odd. Logarithmic function - a general functionA.

6 . The function has no maximum or minimum points, is continuous in the domain of definition.

The following figure shows a graph of a decreasing logarithmic function - (0

If you construct 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 = 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 definition 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 logarithmic functions in GeoGebra

Logarithmic Function Graphs
1) natural logarithm y = ln (x)
2) decimal logarithm y = log(x)
3) base 2 logarithm y = ld (x)

V. Reinforcing the topic

Using the obtained properties of the logarithmic function, we will solve the following problems:

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=log 2 (x+2) -3 y= log 2 (x) +2

Page 1

The logarithmic function (80) performs the inverse mapping of the entire w plane with a cut into a strip - i / /: i, an infinite-sheeted Riemann surface onto the complete z - plane.

Logarithmic function: y logax, where the base of logarithms a is a positive number not equal to one.

The logarithmic function plays a special role in the design and analysis of algorithms, so it is worth considering in more detail. Because we often deal with analytical results in which the constant factor is omitted, we use log TV notation, omitting the base. Changing the logarithm base only changes the value of the logarithm by a constant factor, however, special meanings of the logarithm base arise in certain contexts.

The logarithmic function is the inverse of the exponential function. Its graph (Fig. 247) is obtained from the graph of the exponential function (with the same base) by bending the drawing along the bisector of the first coordinate angle. The graph of any inverse function is also obtained.

The logarithmic function is then introduced as the inverse of the exponential function. The properties of both functions can be easily derived from these definitions. It was this definition that received the approval of Gauss, who at the same time expressed disagreement with the assessment given to him in the review of the Göttingen Scientific News. At the same time, Gauss approached the issue from a broader point of view than da Cunha. The latter limited himself to considering exponential and logarithmic functions in the real region, while Gauss extended their definition to complex variables.

The logarithmic function y logax is monotonic throughout its entire domain of definition.

The logarithmic function is continuous and differentiable throughout its entire domain of definition.

A logarithmic function increases monotonically if a I. For 0 a 1, a logarithmic function with base a decreases monotonically.

The logarithmic function is defined only for positive values ​​of x and one-to-one displays the interval (0; 4 - os.

The logarithmic function y loga x is the inverse function of the exponential function yax.

Logarithmic function: y ogax, where the base of logarithms a is a positive number not equal to one.

Logarithmic functions combine well with physical concepts of the nature of polyethylene creep under conditions where the strain rate is low. In this respect they coincide with the Andraade equation, so they are sometimes used to approximate experimental data.

The logarithmic function, or natural logarithm, and In z, is determined by solving the transcendental equation g ei with respect to u. In the region of real values ​​of x and y, under the condition x 0, this equation admits a unique solution.

“Logarithmic function, its properties and graph.”

Algebra 10th grade

Lesson topic: “Logarithmic function, its properties and graph.”

Lesson type: learning new material.

Lesson objectives:

• 
  form a representation of the logarithmic function and its basic properties;
• 
  develop the ability to plot a logarithmic function;
• 
  promote the development of skills to identify the properties of a logarithmic function from a graph;
• 
  development of skills in working with text, the ability to analyze information, the ability to systematize, evaluate, and use it;
• 
  development of skills to work in pairs, microgroups (communication skills, dialogue, joint decision-making)

Techniques used: true, false statements, INSERT, cluster, syncwine

Equipment: PowerPoint presentation, interactive whiteboard, handouts (cards, text material, tables), squared sheets of paper,

During the classes:

Call stage:

Teacher introduction. We are working on mastering the topic “Logarithms”. What do we currently know and can do?

Student answers.

We know: definition, properties of the logarithm, the basic logarithmic identity, formulas for transition to a new base, areas of application of logarithms.

We can: calculate logarithms, solve simple logarithmic equations, transform logarithms.
What concept is closely related to the concept of logarithm? (with the concept of degree, since logarithm is an exponent)

Student assignment. Using the concept of logarithm, fill in any two tables with

a > 1 and at 0 a (Appendix No. 1)

What do the presented expressions represent? (exponential equations, exponential functions)

Student assignment. Solve exponential equations using variable expression X via variable at.

As a result of this work, the following formulas are obtained:

Let us swap places in the resulting expressions X And at. What did we get?

What would you call these functions? (logarithmic, since the variable is under the logarithm sign). 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 study graphs of logarithmic functions and apply their properties.

You have cards with questions on your 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. Operating time 3 minutes. (Appendix No. 2)

After hearing the students' answers, the first column of the summary table on the board is filled in.
Content comprehension stage(10 min).

By summing up the work with the questions in the table, the teacher prepares students for the idea that when answering questions, we do not yet know whether we are right or wrong.

Group assignment. Answers to the questions can be found by studying the text §4 pp. 240-242. But I suggest not just reading the text, but choosing one of the four previously obtained functions: , , , , constructing its graph and identifying the properties of the logarithmic function from the graph. Each group member does this in a notebook. And then a graph of the function is built on a large sheet of paper in a square. After completion of the work, a representative of each group speaks in defense of their work.
Group assignment. Generalize the properties of the function 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
Graph of a function y = log a x passes through a point with coordinates (1;0)

Group assignment. Prove that the exponential and logarithmic functions are mutually inverse.

Students draw a graph of a logarithmic and exponential function in the same coordinate system

Let us consider two functions simultaneously: exponential y = a X and logarithmic y = log a X.

Figure 2 schematically shows the graphs of the functions y = a x And y = log a X in case a>1.

Figure 3 schematically shows the graphs of the functions y = a x And y = log a X in case 0
Fig.3.

The following statements are true.

• 
  Graph of a function y = log a X is symmetrical to the graph of the function y = a x relative to the straight line y = x.
• 
  Function value set y = a x is a set y>0, and the domain of definition of the function y = log a X is a 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 0у = log a X also decreases at 0

Therefore, indicative y = a x and logarithmic y = log a X the functions are mutually inverse.
Graph of a function y = log a X called a logarithmic curve, although in fact a new name could not be invented. After all, this is the same exponent that serves as the graph of the exponential function, only located differently on the coordinate plane.
Reflection stage. Preliminary summary.

Let's return to the questions discussed at the beginning of the lesson and discuss the results obtained. Let's see, maybe our opinion has changed after work.

Students in groups compare their assumptions with the information obtained from working with the textbook, constructing graphs of functions and descriptions of their properties, make changes to the table, share their thoughts with the class, and discuss the answers to each question.

Call stage. In what cases do you think, when performing what tasks can the properties of a logarithmic function be applied?

Expected 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)

1. Find the domain of the function:

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

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

A) x≠0 b) x>0 V) .

Homework:§ 4 p.240-243, No. 69-75 (even)

Literature:

1. 
   Azevich A.I. Twenty lessons of harmony: Humanities and mathematics course. - M.: Shkola-Press, 1998.-160 p.: ill. (Library of the journal “Mathematics at School”. Issue 7.)
2. 
   Zaire.Bek S.I. Development of critical thinking in the classroom: a manual for general education teachers. institutions. – M. Education, 2011. – 223 p.
3. 
   Kolyagin Yu.M. Algebra and the beginnings of analysis. 10th grade: textbook. for general education institutions: basic and profile levels. – M.: Education, 2010.
4. 
   Korchagin V.V. Unified State Exam 2009. Mathematics. Thematic training tasks. – M.: Eksmo, 2009.
5. 
   Unified State Exam 2008. Mathematics. Thematic training tasks/ Koreshkova T.A. and others. - M.: Eksmo, 2008

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.

Private values

The base 10 logarithm 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

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.