Exponents equations and inequalities examples. Exponential equations and inequalities

In this lesson we will look at various exponential inequalities and learn how to solve them, based on the technique for solving the simplest exponential inequalities

1. Definition and properties of an exponential function

Let us recall the definition and basic properties of the exponential function. The solution of all exponential equations and inequalities is based on these properties.

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

Rice. 1. Graph of exponential function

The graph shows increasing and decreasing exponents, illustrating the exponential function with 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 with, decreases with.

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

When , when the argument increases from minus to plus infinity, the function increases from zero inclusive to plus infinity, i.e., for given values ​​of the argument we have a monotonically increasing function (). On the contrary, when the argument increases from minus to plus infinity, the function decreases from infinity to zero inclusive, i.e., for given values ​​of the argument we have a monotonically decreasing function ().

2. The simplest exponential inequalities, solution method, example

Based on the above, we present a method for solving simple exponential inequalities:

Technique for solving inequalities:

Equalize the bases of degrees;

Compare metrics by saving or changing to opposite sign inequalities.

The solution to complex exponential inequalities usually consists in reducing them to the simplest exponential inequalities.

The base of the degree is greater than one, which means the inequality sign is preserved:

Let's transform right side according to the properties of the degree:

The base of the degree is less than one, the inequality sign must be reversed:

To solve the quadratic inequality, we solve the corresponding quadratic equation:

Using Vieta's theorem we find the roots:

The branches of the parabola are directed upward.

Thus, we have a solution to the inequality:

It’s easy to guess that the right side can be represented as a power with an exponent of zero:

The base of the degree is greater than one, the inequality sign does not change, we get:

Let us recall the technique for solving such inequalities.

Consider the fractional-rational function:

We find the domain of definition:

Finding the roots of the function:

The function has a single root,

We select intervals of constant sign and determine the signs of the function on each interval:

Rice. 2. Intervals of constancy of sign

Thus, we received the answer.

Answer:

3. Solving standard exponential inequalities

Let's consider inequalities with the same indicators, but different bases.

One of the properties of the exponential function is that for any value of the argument it takes strictly positive values, which means it can be divided into an exponential function. Let us divide the given inequality by its right side:

The base of the degree is greater than one, the inequality sign is preserved.

Let's illustrate the solution:

Figure 6.3 shows graphs of functions and . Obviously, when the argument is greater than zero, the graph of the function is higher, this function is larger. When the argument values ​​are negative, the function goes lower, it is smaller. If the argument is equal, the functions are equal, which means that this point is also a solution to the given inequality.

Rice. 3. Illustration for example 4

Let us transform the given inequality according to the properties of the degree:

Here are some similar terms:

Let's divide both parts into:

Now we continue to solve similarly to example 4, divide both parts by:

The base of the degree is greater than one, the inequality sign remains:

4. Graphical solution of exponential inequalities

Example 6 - Solve the inequality graphically:

Let's look at the functions on the left and right sides and build a graph for each of them.

The function is exponential and increases over its entire domain of definition, i.e., for all real values ​​of the argument.

The function is linear and decreases over its entire domain of definition, i.e., for all real values ​​of the argument.

If these functions intersect, that is, the system has a solution, then such a solution is unique and can be easily guessed. To do this, we iterate over integers ()

It is easy to see that the root of this system is:

Thus, the graphs of the functions intersect at a point with an argument equal to one.

Now we need to get an answer. The meaning of the given inequality is that the exponent must be greater than or equal to linear function, that is, to be higher or coincide with it. The answer is obvious: (Figure 6.4)

Rice. 4. Illustration for example 6

So, we looked at solving various standard exponential inequalities. Next we move on to consider more complex exponential inequalities.

Bibliography

Mordkovich A. G. Algebra and the beginnings of mathematical analysis. - M.: Mnemosyne. Muravin G. K., Muravin O. V. Algebra and the beginnings of mathematical analysis. - M.: Bustard. Kolmogorov A. N., Abramov A. M., Dudnitsyn Yu. P. et al. Algebra and the beginnings of mathematical analysis. - M.: Enlightenment.

Math. md. Mathematics-repetition. com. Diffur. kemsu. ru.

Homework

1. Algebra and the beginnings of analysis, grades 10-11 (A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn) 1990, No. 472, 473;

2. Solve the inequality:

3. Solve inequality.

Belgorod State University

DEPARTMENT algebra, number theory and geometry

Work theme: Exponential power equations and inequalities.

Graduate work student of the Faculty of Physics and Mathematics

Scientific adviser:

______________________________

Reviewer: _______________________________

________________________

Belgorod. 2006

Introduction.

“...the joy of seeing and understanding...”

A. Einstein.

In this work, I tried to convey my experience of working as a mathematics teacher, to convey at least to some extent my attitude towards its teaching - a human endeavor in which both are surprisingly intertwined. mathematical science, and pedagogy, and didactics, and psychology, and even philosophy.

I had the opportunity to work with kids and graduates, with children standing at the poles intellectual development: those who were registered with a psychiatrist and who were really interested in mathematics

I had the opportunity to solve many methodological problems. I will try to talk about those that I managed to solve. But even more failed, and even in those that seem to have been resolved, new questions arise.

But even more important than the experience itself are the teacher’s reflections and doubts: why is it exactly like this, this experience?

And the summer is different now, and the development of education has become more interesting. “Under the Jupiters” is now not a search for mythical optimal system teaching “everyone and everything”, but the child himself. But then - of necessity - the teacher.

IN school course algebra and beginning of analysis, grades 10 - 11, with passing the Unified State Exam per course high school and on entrance exams to universities there are equations and inequalities containing an unknown in the base and exponents - these are exponential equations and inequalities.

They receive little attention at school; there are practically no assignments on this topic in textbooks. However, mastering the methodology for solving them, it seems to me, is very useful: it increases the mental and creative abilities of students, and completely new horizons open up before us. When solving problems, students acquire first skills research work, their mathematical culture is enriched, their abilities to logical thinking. Schoolchildren develop such personality qualities as determination, goal-setting, and independence, which will be useful to them in later life. And also there is repetition, expansion and deep assimilation of educational material.

I started working on this topic for my thesis by writing my coursework. In the course of which I deeply studied and analyzed the mathematical literature on this topic, I identified the most suitable method for solving exponential equations and inequalities.

It lies in the fact that in addition to the generally accepted approach when solving exponential equations (the base is taken greater than 0) and when solving the same inequalities (the base is taken greater than 1 or greater than 0, but less than 1), cases are also considered when the bases are negative, equal 0 and 1.

An analysis of students’ written examination papers shows that the lack of coverage of the question of negative value argument of the exponential function in school textbooks causes them a number of difficulties and leads to errors. And they also have problems at the stage of systematizing the results obtained, where, due to the transition to an equation - a consequence or an inequality - a consequence, extraneous roots may appear. In order to eliminate errors, we use a test using the original equation or inequality and an algorithm for solving exponential equations, or a plan for solving exponential inequalities.

In order for students to successfully pass final and entrance exams, I believe it is necessary to pay more attention to solving exponential equations and inequalities in training sessions, or additionally in electives and clubs.

Thus subject , my thesis is defined as follows: “Exponential power equations and inequalities.”

Goals of this work are:

1. Analyze the literature on this topic.

2. Give full analysis solving exponential power equations and inequalities.

3. Provide a sufficient number of examples of various types on this topic.

4. Check in class, elective and club classes how the proposed methods for solving exponential equations and inequalities will be perceived. Give appropriate recommendations for studying this topic.

Subject Our research is to develop a methodology for solving exponential equations and inequalities.

The purpose and subject of the study required solving the following problems:

1. Study the literature on the topic: “Exponential power equations and inequalities.”

2. Master the techniques for solving exponential equations and inequalities.

3. Select training material and develop a system of exercises at different levels on the topic: “Solving exponential equations and inequalities.”

During the thesis research, more than 20 works devoted to the use of various methods solving exponential power equations and inequalities. From here we get.

Thesis plan:

Introduction.

Chapter I. Analysis of literature on the research topic.

Chapter II. Functions and their properties used in solving exponential equations and inequalities.

II.1. Power function and its properties.

II.2. Exponential function and its properties.

Chapter III. Solving exponential power equations, algorithm and examples.

Chapter IV. Solving exponential inequalities, solution plan and examples.

Chapter V. Experience of conducting classes with schoolchildren on this topic.

1.Training material.

2.Tasks for independent solution.

Conclusion. Conclusions and offers.

List of used literature.

Chapter I analyzes the literature

Exponential equations and inequalities are those in which the unknown is contained in the exponent.

Solving exponential equations often comes down to solving the equation a x = a b, where a > 0, a ≠ 1, x is an unknown. This equation has a single root x = b, since the following theorem is true:

Theorem. If a > 0, a ≠ 1 and a x 1 = a x 2, then x 1 = x 2.

Let us substantiate the considered statement.

Let us assume that the equality x 1 = x 2 does not hold, i.e. x 1< х 2 или х 1 = х 2 . Пусть, например, х 1 < х 2 . Тогда если а >1, then the exponential function y = a x increases and therefore the inequality a x 1 must be satisfied< а х 2 ; если 0 < а < 1, то функция убывает и должно выполняться неравенство а х 1 >a x 2. In both cases we received a contradiction to the condition a x 1 = a x 2.

Let's consider several problems.

Solve the equation 4 ∙ 2 x = 1.

Solution.

Let's write the equation in the form 2 2 ∙ 2 x = 2 0 – 2 x+2 = 2 0, from which we get x + 2 = 0, i.e. x = -2.

Answer. x = -2.

Solve equation 2 3x ∙ 3 x = 576.

Solution.

Since 2 3x = (2 3) x = 8 x, 576 = 24 2, the equation can be written as 8 x ∙ 3 x = 24 2 or as 24 x = 24 2.

From here we get x = 2.

Answer. x = 2.

Solve the equation 3 x+1 – 2∙3 x - 2 = 25.

Solution.

Taking the common factor 3 x - 2 out of brackets on the left side, we get 3 x - 2 ∙ (3 3 – 2) = 25 – 3 x - 2 ∙ 25 = 25,

whence 3 x - 2 = 1, i.e. x – 2 = 0, x = 2.

Answer. x = 2.

Solve the equation 3 x = 7 x.

Solution.

Since 7 x ≠ 0, the equation can be written as 3 x /7 x = 1, whence (3/7) x = 1, x = 0.

Answer. x = 0.

Solve the equation 9 x – 4 ∙ 3 x – 45 = 0.

Solution.

By replacing 3 x = a, this equation is reduced to the quadratic equation a 2 – 4a – 45 = 0.

Solving this equation, we find its roots: a 1 = 9, and 2 = -5, whence 3 x = 9, 3 x = -5.

The equation 3 x = 9 has root 2, and the equation 3 x = -5 has no roots, since the exponential function cannot take negative values.

Answer. x = 2.

Solving exponential inequalities often comes down to solving the inequalities a x > a b or a x< а b . Эти неравенства решаются с помощью свойства возрастания или убывания показательной функции.

Let's look at some problems.

Solve inequality 3 x< 81.

Solution.

Let's write the inequality in the form 3 x< 3 4 . Так как 3 >1, then the function y = 3 x is increasing.

Therefore, for x< 4 выполняется неравенство 3 х < 3 4 , а при х ≥ 4 выполняется неравенство 3 х ≥ 3 4 .

Thus, at x< 4 неравенство 3 х < 3 4 является верным, а при х ≥ 4 – неверным, т.е. неравенство
3 x< 81 выполняется тогда и только тогда, когда х < 4.

Answer. X< 4.

Solve the inequality 16 x +4 x – 2 > 0.

Solution.

Let us denote 4 x = t, then we obtain the quadratic inequality t2 + t – 2 > 0.

This inequality holds for t< -2 и при t > 1.

Since t = 4 x, we get two inequalities 4 x< -2, 4 х > 1.

The first inequality has no solutions, since 4 x > 0 for all x € R.

We write the second inequality in the form 4 x > 4 0, whence x > 0.

Answer. x > 0.

Graphically solve the equation (1/3) x = x – 2/3.

Solution.

1) Let's build graphs of the functions y = (1/3) x and y = x – 2/3.

2) Based on our figure, we can conclude that the graphs of the considered functions intersect at the point with the abscissa x ≈ 1. Checking proves that

x = 1 is the root of this equation:

(1/3) 1 = 1/3 and 1 – 2/3 = 1/3.

In other words, we have found one of the roots of the equation.

3) Let's find other roots or prove that there are none. The function (1/3) x is decreasing, and the function y = x – 2/3 is increasing. Therefore, for x > 1, the values ​​of the first function are less than 1/3, and the second – more than 1/3; at x< 1, наоборот, значения первой функции больше 1/3, а второй – меньше 1/3. Геометрически это означает, что графики этих функций при х >1 and x< 1 «расходятся» и потому не могут иметь точек пересечения при х ≠ 1.

Answer. x = 1.

Note that from the solution of this problem, in particular, it follows that the inequality (1/3) x > x – 2/3 is satisfied for x< 1, а неравенство (1/3) х < х – 2/3 – при х > 1.

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and x = b is the simplest exponential equation. In him a greater than zero and A does not equal one.

Solving exponential equations

From the properties of the exponential function we know that its range of values ​​is limited to positive real numbers. Then if b = 0, the equation has no solutions. The same situation occurs in the equation where b

Now let us assume that b>0. If in the exponential function the base a is greater than unity, then the function will be increasing over the entire domain of definition. If in the exponential function for the base A done next condition 0

Based on this and applying the root theorem, we find that the equation a x = b has one single root, for b>0 and positive a not equal to one. To find it, you need to represent b as b = a c.
Then it is obvious that With will be a solution to the equation a x = a c .

Consider the following example: solve the equation 5 (x 2 - 2*x - 1) = 25.

Let's imagine 25 as 5 2, we get:

5 (x 2 - 2*x - 1) = 5 2 .

Or what is equivalent:

x 2 - 2*x - 1 = 2.

We solve the resulting quadratic equation by any of known methods. We get two roots x = 3 and x = -1.

Answer: 3;-1.

Let's solve the equation 4 x - 5*2 x + 4 = 0. Let's make the replacement: t=2 x and get the following quadratic equation:

t 2 - 5*t + 4 = 0.
We solve this equation using any of the known methods. We get the roots t1 = 1 t2 = 4

Now we solve the equations 2 x = 1 and 2 x = 4.

Answer: 0;2.

Solving exponential inequalities

The solution to the simplest exponential inequalities is also based on the properties of increasing and decreasing functions. If in an exponential function the base a is greater than one, then the function will be increasing over the entire domain of definition. If in the exponential function for the base A the following condition is met 0, then this function will be decreasing on the entire set of real numbers.