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A solution method is good if from the very beginning we can foresee - and subsequently confirm this -
that by following this method, we will reach the goal.
G. Leibniz

LESSON TYPE: Consolidation and improvement of knowledge.

• Didactic - Repeat and consolidate the properties of logarithms; logarithmic equations; fix methods for solving the largest and smallest values ​​of a function; improve the application of the acquired knowledge in solving problems of the Unified State Examination C1 and C3;
• Educational - Development of logical thinking, memory, cognitive interest, continue the formation of mathematical speech and graphic culture, develop the ability to analyze;
• Educational - To accustom to the aesthetic design of notes in a notebook, the ability to communicate, to instill accuracy.

Equipment: blackboard, computer, projector, screen, cards with test tasks, with tasks for the work of all students.

Forms of work: f oral, individual, collective.

DURING THE CLASSES

1. ORGANIZATIONAL TIME

2. GOAL SETTING

3. CHECK HOMEWORK

4. UPDATED KNOWLEDGE

Analyze: in which tasks of the exam there are logarithms.

(V-7 simplest logarithmic equations

B-11-transformation of logarithmic expressions

B-12 - problems of physical content related to logarithms

B-15- Finding the largest and smallest value of a function

C-1 - trigonometric equations containing a logarithm

C-3 - a system of inequalities containing a logarithmic inequality)

At this stage, oral work is carried out, during which students not only remember the properties of logarithms, but also perform the simplest tasks of the exam.

1) Definition of the logarithm. What properties of the logarithm do you know? (and conditions?)

1. log b b = 1
2. log b 1 = 0, 3. log c (ab) = log c a + log c b.
4. log c (a: b) = log c a - log c b.
5. log c (b k) = k * log c

2) What is the logarithmic function? D(y) -?

3) What is a decimal logarithm? ()

4) What is the natural logarithm? ()

5) What is the number e?

6) What is the derivative of ? ()

7) What is the derivative of the natural logarithm?

5. ORAL WORK for all students

Calculate orally: (tasks B-11)

= = = =

152 1 144 -1/2

6. Independent activity of students in solving tasks

B-7 followed by verification

Solve the equations (the first two equations are spoken orally, and the rest are solved by the whole class on their own and write the solution in a notebook):

(While the students are working on the spot on their own, 3 students come to the board and work on individual cards)

After checking 3-5 equations from the spot, the guys are invited to prove that the equation has no solution (orally)

7. Solution B-12 - (problems of physical content related to logarithms)

The whole class solves the problem (there are 2 people at the board: the 1st solves it together with the class, the 2nd solves a similar problem on their own)

8. ORAL WORK (questions)

Recall the algorithm for finding the largest and smallest values ​​of a function on a segment and on an interval.

Work on the board and in a notebook.

(prototype B15 - USE)

9. Mini-test with self-control.

№	1 option	Option 2
1.	=
2.
3.
4.
5.
6.	Find the largest value of a function 11. The performance of students in the role of experts The guys are invited to evaluate the student's work - task C-1, completed on the examination form - 0.1.2 points (see presentation) 12. HOMEWORK The teacher explains the homework, paying attention to the fact that similar tasks were considered in the lesson. Students listen carefully to the explanations of the teacher, write down their homework. FIPI (open bank of tasks: geometry section, 6th page) uztest.ru (transformation of logarithms) C3 - task of the second part of the exam 13. SUMMARY Today in the lesson we repeated the properties of logarithms; logarithmic equations; fixed methods for finding the largest and smallest values ​​of a function; considered the problems of physical content related to logarithms; solved problems C1 and C3, which are offered at the exam in mathematics in prototypes B7, B11, B12, B15, C1 and C3. Grading.

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How to solve the USE problem No. 13 for exponential and logarithmic equations | 1C: Tutor

What do you need to know about exponential and logarithmic equations for solving USE problems in mathematics?

Being able to solve exponential and logarithmic equations is very important for the successful passing of the unified state exam in mathematics at the profile level. Important for two reasons:

First of all, task No. 13 of the KIM USE variant, albeit infrequently, but still sometimes it is just such an equation that you need not only to solve, but also (similar to the trigonometry task) to choose the roots of the equation that satisfy any condition.

So, one of the options for 2017 included the following task:

a) Solve the equation 8 x – 7 . 4 x – 2 x +4 + 112 = 0.

b) Indicate the roots of this equation that belong to the segment.

Answer: a) 2; log 2 7 and b) log 2 7.

In another version, there was such a task:

a) Solve the equation 6log 8 2 x– 5 log 8 x + 1 = 0

b) Find all the roots of this equation that belong to the segment.

Answer: a) 2 and 2√ 2 ; b) 2.

There was also this:

a) Solve the equation 2log 3 2 (2cos x) – 5log 3 (2cos x) + 2 = 0.

b) Find all the roots of this equation that belong to the segment [π; 5π/2].

Answer: a) (π/6 + 2πk; -π/6 + 2πk, k∊Z) and b) 11π/6; 13π/6.

Secondly, the study of methods for solving exponential and logarithmic equations is good, since the basic methods for solving both equations and inequalities actually use the same mathematical ideas.

The main methods for solving exponential and logarithmic equations are easy to remember, there are only five of them: reduction to the simplest equation, the use of equivalent transitions, the introduction of new unknowns, the logarithm and factorization. Separately, there is a method of using the properties of exponential, logarithmic and other functions in solving problems: sometimes the key to solving an equation is the domain of definition, the range of values, non-negativity, boundedness, evenness of the functions included in it.

As a rule, in problem No. 13 there are equations that require the use of the five main methods listed above. Each of these methods has its own characteristics that you need to know, since it is their ignorance that leads to errors in solving problems.

What are the common mistakes examiners make?

Often, when solving equations containing an exponential-power function, students forget to consider one of the cases where the equality is satisfied. As is well known, equations of this form are equivalent to a set of two systems of conditions (see below), we are talking about the case when a( x) = 1

This error is due to the fact that when solving the equation, the examinee formally uses the definition of the exponential function (y= ax, a>0, a ≠ 1): at a ≤ 0 exponential function is not really defined,

But at a = 1 is defined, but is not exponential, since the unit in any real power is identically equal to itself. This means that if in the considered equation at a(x) = 1 there is a true numerical equality, then the corresponding values ​​of the variable will be the roots of the equation.

Another mistake is applying the properties of logarithms without taking into account the range of acceptable values. For example, the well-known property “the logarithm of a product is equal to the sum of logarithms” turns out to have a generalization:
log a( f(x)g(x)) = log a │ f(x)│ + log a │g( x)│, at f(x)g(x) > 0, a > 0, a ≠ 1

Indeed, in order for the expression on the left side of this equality to be defined, it is sufficient that the product of the functions f and g was positive, but the functions themselves can be both greater and less than zero at the same time, therefore, when applying this property, it is necessary to use the concept of a module.

And there are many such examples. Therefore, for the effective development of methods for solving exponential and logarithmic equations, it is best to use the services that will be able to talk about such "pitfalls" using examples of solving the corresponding examination problems.

Practice problem solving regularly

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In task No. 12 of the Unified State Examination in mathematics of the profile level, we need to find the largest or smallest value of the function. To do this, it is necessary to use, obviously, the derivative. Let's look at a typical example.

Analysis of typical options for assignments No. 12 USE in mathematics at the profile level

The first version of the task (demo version 2018)

Find the maximum point of the function y = ln(x+4) 2 +2x+7.

Solution algorithm:

1. We find the derivative.
2. We write down the answer.

Decision:

1. We are looking for x values ​​for which the logarithm makes sense. To do this, we solve the inequality:

Since the square of any number is non-negative. The only solution to the inequality is the value of x for which x + 4≠ 0, i.e. at x≠-4.

2. Find the derivative:

y'=(ln(x+4) 2 + 2x + 7)'

By the property of the logarithm, we get:

y'=(ln(x+4) 2)'+(2x)'+(7)'.

According to the formula for the derivative of a complex function:

(lnf)'=(1/f)∙f'. We have f=(x+4) 2

y, = (ln(x+4) 2)'+ 2 + 0 = (1/(x+4) 2)∙((x+4) 2)' + 2=(1/(x+4) 2 2) ∙ (x 2 + 8x + 16) ' + 2 \u003d 2 (x + 4) / ((x + 4) 2) + 2

y'= 2/(x + 4) + 2

3. Equate the derivative to zero:

y, = 0 → (2+2∙(x + 4))/(x + 4)=0,

2 + 2x +8 = 0, 2x + 10 = 0,

The second version of the task (from Yaschenko, No. 1)

Find the minimum point of the function y = x - ln(x+6) + 3.

Solution algorithm:

1. We define the scope of the function.
2. We find the derivative.
3. We determine at what points the derivative is equal to 0.
4. We exclude points that do not belong to the domain of definition.
5. Among the remaining points, we are looking for x values ​​at which the function has a minimum.
6. We write down the answer.

Decision:

1. ODZ:.

2. Find the derivative of the function:

3. Equate the resulting expression to zero:

4. We got one point x=-5, which belongs to the domain of the function.

5. At this point, the function has an extremum. Let's see if this is the minimum. At x=-4

At x = -5.5, the derivative of the function is negative, since

Hence, the point x=-5 is the minimum point.

The third version of the task (from Yaschenko, No. 12)

Solution algorithm:.

1. We find the derivative.
2. We determine at what points the derivative is equal to 0.
3. We exclude points that do not belong to a given segment.
4. Among the remaining points, we are looking for the x values ​​at which the function has a maximum.
5. We find the values ​​of the function at the ends of the segment.
6. We are looking for the largest among the obtained values.
7. We write down the answer.

Decision:

1. We calculate the derivative of the function, we get