GBOU secondary school No. 149 of St. Petersburg

Lesson summary

Novikova Olga Nikolaevna

2016

Topic: "System of exponential equations and inequalities".

Lesson Objectives:

educational:

generalize and consolidate knowledge about how to solve exponential equations and inequalities contained in systems of equations and inequalities

developing: activation cognitive activity; development of skills of self-control and self-assessment, self-analysis of their activities.

educational: formation of skills to work independently; make decisions and draw conclusions; education of aspiration to self-education and self-improvement.

Lesson type : combined.

Type of lesson: practical lesson.

During the classes

I. Organizing time(1 minute)

Formulation of the goal for the class: Generalize and consolidate knowledge about how to solve exponential equations and inequalities contained in systems of equations and inequalities based on the properties of the exponential function.

II. Oral work (1 minute)

Definition of an exponential equation.
Methods for solving exponential equations.
Algorithm for solving exponential inequalities.

III . Examination homework(3 min)

Students in their places. The teacher checks the answers and asks how to solve demonstrative equations and inequalities. №228-231(odd)

IV. Updating of basic knowledge. "Brainstorm": (3 min)

Questions are shown printed sheets on the students' desks "Exponential functions, equations, inequalities" and are offered to students for oral answers from the spot.

1. What function is called exponential?

2. What is the scope of the function y= 0,5x?

3. What is the domain of the exponential function?

4. What is the scope of the function y= 0,5x?

5. What properties can a function have?

6. Under what condition is the exponential function increasing?

7. Under what condition is the exponential function decreasing?

8. Increasing or decreasing exponential function

9. What equation is called exponential?

Diagnostics of the level of formation of practical skills.

Task 10 write down the solution in notebooks. (7 min)

10. Knowing the properties of an increasing and decreasing exponential function, solve the inequalities

2 3 < 2 X ;
; 3 X < 81 ; 3 X < 3 4

11 . Solve the equation: 3 x = 1

12 . Calculate 7.8 0 ; 9.8 0

13 . Specify a method for solving exponential equations and solve it:

After completion, the pairs change leaves. I appreciate each other. Criteria on the board. Checking against records on sheets in a file.

Thus, we repeated the properties of the exponential function, methods for solving exponential equations.

The teacher selectively takes and evaluates the work of 2-3 students.

Solution Workshop systems exponential equations and inequalities: (23 min)

Consider the solution of systems of exponential equations and inequalities based on the properties of the exponential function.

When solving systems of exponential equations and inequalities, the same techniques are used as when solving systems of algebraic equations and inequalities (substitution method, addition method, method of introducing new variables). In many cases, before applying one or another solution method, it is necessary to transform each equation (inequality) of the system to the simplest possible form.

Examples.

1.

Solution:

Answer: (-7; 3); (1; -1).

2.

Solution:

Denote 2 X= u, 3 y= v. Then the system will be written like this:

Let's solve this system using the substitution method:

Equation 2 X= -2 has no solutions, because -2<0, а 2 X> 0.

b)

Answer: (2;1).

№244(1)

Answer: 1.5; 2

Summarizing. Reflection. (5 minutes)

Lesson summary: Today we have repeated and summarized the knowledge of methods for solving exponential equations and inequalities contained in systems based on the properties of the exponential function.

The children in turn are invited to take from the following phrases to choose and continue the phrase.

Reflection:

today I found out...

it was difficult…

I understand that…

I have learned...

I could)…

It was interesting to know that...

surprised me...

I wanted…

Homework. (2 minutes)

No. 240-242 (odd) p.86

In this lesson, we will consider the solution of more complex exponential equations, recall the main theoretical provisions regarding the exponential function.

1. Definition and properties of an exponential function, a technique for solving the simplest exponential equations

Recall the definition and main properties of an exponential function. It is on the properties that the solution of all exponential equations and inequalities is based.

Exponential function is a function of the form , where the base is the degree and Here x is an independent variable, an argument; y - dependent variable, function.

Rice. 1. Graph of the exponential function

The graph shows an increasing and decreasing exponent, illustrating the exponential function at a base greater than one and less than one, but greater than zero, respectively.

Both curves pass through the point (0;1)

Properties of the exponential function:

Domain: ;

Range of values: ;

The function is monotonic, increases as , decreases as .

A monotonic function takes each of its values ​​with a single value of the argument.

When the argument increases from minus to plus infinity, the function increases from zero, inclusive, to plus infinity. On the contrary, when the argument increases from minus to plus infinity, the function decreases from infinity to zero, inclusive.

2. Solution of typical exponential equations

Recall how to solve the simplest exponential equations. Their solution is based on the monotonicity of the exponential function. Almost all complex exponential equations are reduced to such equations.

The equality of exponents with equal bases is due to the property of the exponential function, namely its monotonicity.

Solution Method:

Equalize the bases of the degrees;

Equate exponents.

Let's move on to more complex exponential equations, our goal is to reduce each of them to the simplest.

Let's get rid of the root on the left side and reduce the degrees to the same base:

In order to reduce a complex exponential equation to a simple one, a change of variables is often used.

Let's use the degree property:

We introduce a replacement. Let then

We multiply the resulting equation by two and transfer all the terms to the left side:

The first root does not satisfy the interval of y values, we discard it. We get:

Let's bring the degrees to the same indicator:

We introduce a replacement:

Let then . With this replacement, it is obvious that y takes strictly positive values. We get:

We know how to solve similar quadratic equations, we write out the answer:

To make sure that the roots are found correctly, you can check according to the Vieta theorem, that is, find the sum of the roots and their product and check with the corresponding coefficients of the equation.

We get:

3. Technique for solving homogeneous exponential equations of the second degree

Let us study the following important type of exponential equations:

Equations of this type are called homogeneous of the second degree with respect to the functions f and g. On its left side there is a square trinomial with respect to f with parameter g or a square trinomial with respect to g with parameter f.

Solution Method:

This equation can be solved as a quadratic one, but it is easier to do it the other way around. Two cases should be considered:

In the first case, we get

In the second case, we have the right to divide by the highest degree and we get:

You should introduce a change of variables , we get a quadratic equation for y:

Note that the functions f and g can be arbitrary, but we are interested in the case when these are exponential functions.

4. Examples of solving homogeneous equations

Let's move all the terms to the left side of the equation:

Since the exponential functions acquire strictly positive values, we have the right to immediately divide the equation by , without considering the case when:

We get:

We introduce a replacement: (according to the properties of the exponential function)

We got a quadratic equation:

We determine the roots according to the Vieta theorem:

The first root does not satisfy the interval of y values, we discard it, we get:

Let's use the properties of the degree and reduce all degrees to simple bases:

It is easy to notice the functions f and g:

Ways to solve systems of equations

To begin with, let us briefly recall what methods of solving systems of equations generally exist.

Exist four main ways solutions of systems of equations:

Substitution method: take any of these equations and express $y$ in terms of $x$, then $y$ is substituted into the equation of the system, from where the variable $x.$ is found. After that, we can easily calculate the variable $y.$

Addition method: in this method, one or both equations must be multiplied by numbers such that when both are added together, one of the variables “disappears”.

Graphical method: both equations of the system are depicted on coordinate plane and find their point of intersection.

The method of introducing new variables: in this method, we do the replacement of some expressions to simplify the system, and then apply one of the above methods.

Systems of exponential equations

Definition 1

Systems of equations consisting of exponential equations are called a system of exponential equations.

We will consider the solution of systems of exponential equations using examples.

Example 1

Solve a system of equations

Picture 1.

Solution.

We will use the first method to solve this system. First, let's express $y$ in the first equation in terms of $x$.

Figure 2.

Substitute $y$ into the second equation:

\ \ \[-2-x=2\] \ \

Answer: $(-4,6)$.

Example 2

Solve a system of equations

Figure 3

Solution.

This system is equivalent to the system

Figure 4

We apply the fourth method for solving equations. Let $2^x=u\ (u >0)$ and $3^y=v\ (v >0)$, we get:

Figure 5

We solve the resulting system by the addition method. Let's add the equations:

\ \

Then from the second equation, we get that

Returning to the replacement, I received a new system of exponential equations:

Figure 6

We get:

Figure 7

Answer: $(0,1)$.

Systems of exponential inequalities

Definition 2

Systems of inequalities consisting of exponential equations are called a system of exponential inequalities.

We will consider the solution of systems of exponential inequalities using examples.

Example 3

Solve the system of inequalities

Figure 8

Solution:

This system of inequalities is equivalent to the system

Figure 9

To solve the first inequality, recall the following equivalence theorem for exponential inequalities:

Theorem 1. The inequality $a^(f(x)) >a^(\varphi (x)) $, where $a >0,a\ne 1$ is equivalent to the set of two systems

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