CC BY
1UDC 512.13
METHODS OF SOLVING EXPONENTIAL-LOGARITHMIC EQUATIONS IN A SCHOOL MATHEMATICS COURSE
SEISENBAYEVA AIZADA MURATOVNA
Master of Mathematics, Senior lecturer at the Kazakh National Women's Pedagogical
University, Almaty, Kazakhstan
ZHANBYRBAY ASEM GABIDENOVNA
2nd year student of the educational program «6b01502-Mathematics-physics» of the Kazakh National Women's Pedagogical University,
Almaty, Kazakhstan
ABDULLAEVA SARVINOZ TULKUNOVNA 2nd year student of the educational program «6b01502-Mathematics-physics» of the Kazakh National Women's Pedagogical University,
Almaty, Kazakhstan
Abstract. In the Republic of Kazakhstan, positive changes and achievements in modern economic and social conditions require improvement of the country's education system. Along with the traditional methods used in the study of mathematics to achieve education at the level of international standards, in-depth study of mathematics in elective courses is of particular importance. Of course, for this, along with the tasks specified in the state standard, it is necessary to teach students complex tasks. In particular, the role of teaching the solution of exponential logarithmic equations, which are not given much attention in secondary school mathematics, is exceptional. After all, in the organization of education, a mathematics teacher should develop logical thinking in students, provide them with the necessary knowledge, and form their problemsolving skills. Improving the quality of education and linking practice with life were also on the agenda. The only way to solve it is to develop ways and didactic justification for solving exponential-logarithmic equations. By teaching students, they develop and improve scientific and logical thinking for the assimilation of material, the level of creative approach to a particular type of pro
Keywords: exponential, logarithmic, equations, system, solutions.
In recent years, unknown grounds and exponential logarithmic equations have been found in examination questions intended for admission to higher educational institutions. Such equations are called exponential-logarithmic equations.
Since little attention is paid to them at school, there are very few tasks related to this topic in textbooks. It is useful to explore ways to solve exponential logarithmic equations, as new horizons open up before us, and solving such problems increases the mental and creative abilities of students. In the process of solving problems, students gain the earliest experience of research work. Their mathematical cultures are expanding, and their logical thinking abilities are developing.
The article considers the forms of formation of cognitive activity of students in solving exponential logarithmic equations in teaching mathematics; problems of organizing the educational process in solving equations, the algorithm for solving equations and examples in which the same algorithm was used [1, 28-75 p.].
Solving a system of exponential equations.
Let's consider the solution of a system of exponential equations.
Definition. We call a system of equations containing an exponential equation a system of exponential equations [2, 47-48 p.].
To solve the system of exponential equations, the properties of the exponential function, exponential equations and methods for solving systems of equations are used. Let's look at the examples. Example 1.
let's solve the system of equations Decision.
To solve a given system of equations, we apply the substitution approach and obtain a system of equations with values:
From here 4X + 4-3~* = 80 or 4* + — go = 0 the equation is obtained. From here
From here 4X = z, z2 - 80z + 1024 = 0 solving the quadratic equation. The roots of the resulting quadratic equation z1 = 16 and z2 = 64. From here 4* = 16 and 4* = 64.
Answer: (3;2)(2;3). Example 2.
let's solve the system of equations
y= 11
Decision.
Two exponential functions are given in the system of equations. Let's introduce new variables first, that is 2X = u, 3y = v.
Then we get a system of linear equations with two unknowns:
1
By the algebraic method of addition, we multiply the second equation of the last system of equations by 2:
From here 5U = ^ or u = - if we put the value of u- in the first equation of the system, we
From here 2* = - , 3>r = 1 the exponential equations come out. From here x = -2,y = 0. Answer: (-2;0).
Methods for solving exponential-logarithmic equations. Example 1.
Consider the equation x21g2 x = 10 x3 [4-5]. Decision.
0 < x * 1.
If we logarithm both sides of the equation to 10, then we get:
4
4
4
lg x21g21 = lg10 x3; 2lg3 x = 1 + 3lg x;
2lg3 x-3lgx-1 = 0; 2lg3 x + 2-3lgx-3 = 0;
2(lg3 x +1)- 3(lg x +1)= 0;
2(lg x + l)(lg2 x - lg x +1)-3(lg x +1) = 0;
(lg x + 1)(2lg2 x - 2lg x -1)= 0
To find the roots of this equation, you need to determine the solutions of these two equations.
(lgx +1)= 0, (2lg2x-2lgx-1) = 0 (lgx) =-1,(lgx)2 = ^y3 and (lgx\ = ^^ ^
j 1-/3 1+V3
x = —, x = 10 2 and x = 10 2
1 10 2 3
^ 1-V3
Answer: x = —, x = 10 2 , x = 10 2
1 10 2 3
Example 2.
10x21g2 x x3l8x
x3 10
consider the equation.
Decision.
0 < x * 1
x21g2x 1
Let's write the equation as follows -
x3 • x3lg x 100
Then x21g2 x-3lg x-3 = 10 -2
If we logarithm both sides of the equation with base 10, we get:
lg x21g2 x-3lg x-3 = lg10-2;
(2lg2 x - 3lg x - 3)lg x = -2;
2lg3x - 3lg2 x - 3lg x + 2 = 0;
2(lg3 x +1)-3lg x(lg x +1) = 0;
2(lg x + l)(lg2 x - lg x +1)-3lg x(lg x +1) = 0;
(lg x + 1)(2lg2 x - 5lg x + 2)= 0.
Then you need to find (lg x +1) = 0 and (2lg2 x - 5lg x + 2) = 0
lg x = -1, x = —, and lg x = -1, x7 = VÏÔ н lg x = 2, x, = 100 1 10 2 2 3
So, the equation has 3 solutions:
x = —, x = -\/l0, x = 100
1 10 2 3
Answer: x = 1/10,x = Vl0,x = 100 . Example 3.
x2-ig2x-igx2 _1 = Q consider the equation.
x
Decision.
0 < x * 1
let's write the equation in the form x2-g x_lgx = x 1
If we logarithm both sides of the equation with base 10, we get :
lg x2^ x-lg x2 = lg x-1; (2 - lg2 x - lg x2 )lg x = - lg x; (2 - lg2 x - lg x2 )lg x + lg x = 0; lg x(lg2 x + 2lg x - 3)= 0;
To find the roots of this equation, let's solve these equations lgx = 0 and (lg2 x + 2lgx-3)= 0
x = 100 = 1.
lg x = -3, x2 = 10 3 = 0,001 and lg x = 1, x = 101 = 10 . So, the equation has 3 solutions:
x = 10, x = 0,001, x = 10.
Answer: x = 10,x = 0,001,x3 = 10. Example 4.
I |lg2 x-lg x2 | |3
x-1 =x-1
Decision.
0 < x * 1.
To solve this equation, it is necessary to consider the following two cases:
i\ I I i\ Hx -1 >0
1) x -1 = 1; 2) J I
y x -1 * 1
We get two answers from the first equation. xl = 0 0, because does not satisfy the condition, if x2 = 2 satisfied.
And from the system we get the following equality lg2 x - lg x2 = 3 a.
lg2 x - 2lg x - 3 = 0
(lg x\ =-1, x2 = 0,1, (lg x)2 = 3, X3=1000. So, the equation has 3 solutions: x = 2, x2 = 0,1, x = 1000. Answer: x = 2, x2 = 0,1, x = 1000. Example 5. xlg x -1 = 10 (1 - x-g x ) Decision. x>0.
xlg x -1 = 10 -10 x-18 x
x'gx + = 11 x x
xx = t
t * 0.
t2 - 11t +10 = 0
D = (-11)2 - 4 x1x10 = 81
11 ± 9
t =-, t = 1, L = 10
2 1 2
..lg x _ -
1) xlg x = 1, x lg x = x 0 lg X = 0, x1 = 1.
2) x'gx =10, x1®x = xlQgx10, lgx = log 10 = —, lg2 x = 1, lg x = 1, x2 = 10 ;
x lg x
lg x = -1,
x3 = 0,1
So, the equation has 3 solutions:
x = 0,1, x = 1, x = 10.
Answer: x = 0,1, x2 = 1, X = 10.
In this sense, the culture of reporting is considered the end result of knowledge and skills that students must learn. Because, given that the knowledge that a student acquires in mathematics at school is formed when solving a mathematical problem, it can be seen that this is a tool for developing skills necessary for a student's life. It is the individual's ability to solve problems-as a result of knowledge-that reinforces the relevance of the study.
The introduction of solving general exponential power equations and inequalities in high school through the choice component increases the scientific level of the mathematics course, gives a systematic direction to the formation of the most important elements of mathematical culture among students, and also plays an important role in the formation of a dialectical-materialistic approach among students.
Therefore, each graduate should have certain general mathematical knowledge, skills, an understanding of the essence of mathematics in the representation of real-life objects and the formation of a mathematical model of important practical tasks so that general knowledge of mathematics is sufficient for studying other subjects, self-education, and continuing knowledge.
USED LITERATURE
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2. Baryshkin A.G., Shkaliko N.A. About the international competition "Talent Search" // Mathematics at school. 1995. No. 3. - pp. 47-48.
3. Baryshkin A.G., Vakhterov V.P. Selected pedagogical works. M., 1987. - 400 p.
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