Solving exponential power equations. Lecture: “Methods for solving exponential equations

In this lesson we will look at solving more complex exponential equations and recall the basic theoretical principles regarding the exponential function.

1. Definition and properties of the exponential function, methods for solving the simplest exponential equations

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 exponentials, illustrating 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 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, not inclusive.

2. Solving standard exponential equations

Let us remind you how to solve the simplest exponential equations. Their solution is based on the monotonicity of the exponential function. Almost all complex exponential equations can be 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 degrees;

Equate the exponents.

Let's move on to consider 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 bring the degrees to the same base:

In order to reduce a complex exponential equation to its simplest, substitution of variables is often used.

Let's use the power property:

We are introducing a replacement. Let it be then. With such a replacement, it is obvious that y accepts strictly positive values. We get:

Let's multiply the resulting equation by two and move all terms to the left side:

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

Let's reduce the degrees to the same indicator:

Let's introduce a replacement:

Let it be then . With such a replacement, it is obvious that y takes on strictly positive values. We get:

We know how to solve such quadratic equations, we can write down the answer:

To make sure that the roots are found correctly, you can check using Vieta’s theorem, i.e., find the sum of the roots and their product and compare them with the corresponding coefficients of the equation.

We get:

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

Let's 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 the parameter g or a square trinomial with respect to g with the parameter f.

Solution method:

This equation You can solve it as a square, but it’s easier to do it differently. There are two cases to consider:

In the first case we get

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

It is necessary to introduce a change of variables, we obtain a quadratic equation for y:

Let us note that the functions f and g can be any, 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 exponential functions acquire strictly positive values, we have the right to immediately divide the equation by , without considering the case when:

We get:

Let's introduce a replacement: (according to the properties of the exponential function)

We got a quadratic equation:

We determine the roots using Vieta’s theorem:

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

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

It's easy to notice the functions f and g:

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

Exponential equations are those in which the unknown is contained in the exponent. The simplest exponential equation has the form: a x = a b, where a> 0, a 1, x is unknown.

The main properties of powers by which exponential equations are transformed: a>0, b>0.

When solving exponential equations, the following properties of the exponential function are also used: y = a x, a > 0, a1:

To represent a number as a power, use the basic logarithmic identity: b = , a > 0, a1, b > 0.

Problems and tests on the topic "Exponential Equations"

• Exponential equations

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• Exponential equations - Important topics for reviewing the Unified State Examination in mathematics

  Tasks: 14

• Systems of exponential and logarithmic equations - Exponential and logarithmic functions grade 11

  Lessons: 1 Assignments: 15 Tests: 1

• §2.1. Solving exponential equations

  Lessons: 1 Tasks: 27

• §7 Exponential and logarithmic equations and inequalities - Section 5. Exponential and logarithmic functions, grade 10

  Lessons: 1 Tasks: 17

To successfully solve exponential equations, you must know the basic properties of powers, properties of the exponential function, and the basic logarithmic identity.

When solving exponential equations, two main methods are used:

1. transition from the equation a f(x) = a g(x) to the equation f(x) = g(x);
2. introduction of new lines.

Examples.

1. Equations reduced to the simplest. They are solved by reducing both sides of the equation to a power with the same base.

3 x = 9 x – 2.

Solution:

3 x = (3 2) x – 2 ;
3 x = 3 2x – 4 ;
x = 2x –4;
x = 4.

Answer: 4.

2. Equations solved by taking the common factor out of brackets.

Solution:

3 x – 3 x – 2 = 24
3 x – 2 (3 2 – 1) = 24
3 x – 2 × 8 = 24
3 x – 2 = 3
x – 2 = 1
x = 3.

Answer: 3.

3. Equations solved using a change of variable.

Solution:

2 2x + 2 x – 12 = 0
We denote 2 x = y.
y 2 + y – 12 = 0
y 1 = - 4; y2 = 3.
a) 2 x = - 4. The equation has no solutions, because 2 x > 0.
b) 2 x = 3; 2 x = 2 log 2 3 ; x = log 2 3.

Answer: log 2 3.

4. Equations containing powers with two different (not reducible to each other) bases.

3 × 2 x + 1 - 2 × 5 x – 2 = 5 x + 2 x – 2.

3× 2 x + 1 – 2 x – 2 = 5 x – 2 × 5 x – 2
2 x – 2 ×23 = 5 x – 2
×23
2 x – 2 = 5 x – 2
(5/2) x– 2 = 1
x – 2 = 0
x = 2.

Answer: 2.

5. Equations that are homogeneous with respect to a x and b x.

General form: .

9 x + 4 x = 2.5 × 6 x.

Solution:

3 2x – 2.5 × 2 x × 3 x +2 2x = 0 |: 2 2x > 0
(3/2) 2x – 2.5 × (3/2) x + 1 = 0.
Let us denote (3/2) x = y.
y 2 – 2.5y + 1 = 0,
y 1 = 2; y 2 = ½.

Answer: log 3/2 2; - log 3/2 2.

1º. Exponential equations are called equations containing a variable in an exponent.

Solving exponential equations is based on the property of powers: two powers with the same base are equal if and only if their exponents are equal.

2º. Basic methods for solving exponential equations:

1) the simplest equation has a solution;

2) an equation of the form logarithmic to the base a reduce to form ;

3) an equation of the form is equivalent to the equation ;

4) equation of the form is equivalent to the equation.

5) an equation of the form is reduced through substitution to an equation, and then a set of simple exponential equations is solved;

6) equation with reciprocals by substitution they reduce to an equation, and then solve a set of equations;

7) equations homogeneous with respect to a g(x) And b g(x) given that kind through replacement they are reduced to an equation, and then a set of equations is solved.

Classification of exponential equations.

1. Equations solved by going to one base.

Example 18. Solve the equation .

Solution: Let's take advantage of the fact that all bases of powers are powers of the number 5: .

2. Equations solved by passing to one exponent.

These equations are solved by transforming the original equation to the form , which is reduced to its simplest using the property of proportion.

Example 19. Solve the equation:

3. Equations solved by taking the common factor out of brackets.

If each exponent in an equation differs from the other by a certain number, then the equations are solved by putting the exponent with the smallest exponent out of brackets.

Example 20. Solve the equation.

Solution: Let’s take the degree with the smallest exponent out of brackets on the left side of the equation:

Example 21. Solve the equation

Solution: Let's group separately on the left side of the equation the terms containing powers with base 4, on the right side - with base 3, then put the powers with the smallest exponent out of brackets:

4. Equations that reduce to quadratic (or cubic) equations.

The following equations are reduced to a quadratic equation for the new variable y:

a) the type of substitution, in this case;

b) the type of substitution , and .

Example 22. Solve the equation .

Solution: Let's make a change of variable and solve the quadratic equation:

.

Answer: 0; 1.

5. Equations that are homogeneous with respect to exponential functions.

An equation of the form is homogeneous equation second degree relative to unknowns a x And b x. Such equations are reduced by first dividing both sides by and then substituting them into quadratic equations.

Example 23. Solve the equation.

Solution: Divide both sides of the equation by:

Putting , we get a quadratic equation with roots .

Now the problem comes down to solving a set of equations . From the first equation we find that . The second equation has no roots, since for any value x.

Answer: -1/2.

6. Rational equations with respect to exponential functions.

Example 24. Solve the equation.

Solution: Divide the numerator and denominator of the fraction by 3 x and instead of two we get one exponential function:

7. Equations of the form .

Such equations with a set of admissible values ​​(APV), determined by the condition, by taking the logarithm of both sides of the equation are reduced to an equivalent equation, which in turn are equivalent to a set of two equations or.

Example 25. Solve the equation: .

.

Didactic material.

Solve the equations:

1. ; 2. ; 3. ;

4. ; 5. ; 6. ;

9. ; 10. ; 11. ;

14. ; 15. ;

16. ; 17. ;

18. ; 19. ;

20. ; 21. ;

22. ; 23. ;

24. ; 25. .

26. Find the product of the roots of the equation .

27. Find the sum of the roots of the equation .

Find the meaning of the expression:

28. , where x 0- root of the equation ;

29. , where x 0– whole root of the equation .

Solve the equation:

31. ; 32. .

Answers: 10; 2. -2/9; 3. 1/36; 4. 0, 0.5; 50; 6.0; 7. -2; 8.2; 9. 1, 3; 10. 8; 11.5; 12.1; 13. ¼; 14.2; 15. -2, -1; 16. -2, 1; 17.0; 18.1; 19.0; 20. -1, 0; 21. -2, 2; 22. -2, 2; 23.4; 24. -1, 2; 25. -2, -1, 3; 26. -0.3; 27.3; 28.11; 29.54; 30. -1, 0, 2, 3; 31. ; 32. .

Topic No. 8.

Exponential inequalities.

1º. An inequality containing a variable in the exponent is called exponential inequality.

2º. Solution exponential inequalities type is based on the following statements:

if , then the inequality is equivalent to ;

if , then the inequality is equivalent to .

When solving exponential inequalities, the same techniques are used as when solving exponential equations.

Example 26. Solve inequality (method of transition to one base).

Solution: Since , then the given inequality can be written as: . Since , then this inequality is equivalent to the inequality .

Solving the last inequality, we get .

Example 27. Solve the inequality: ( by taking the common factor out of brackets).

Solution: Let's take out of brackets on the left side of the inequality , on the right side of the inequality and divide both sides of the inequality by (-2), changing the sign of the inequality to the opposite:

Since , then when moving to inequality of indicators, the sign of inequality again changes to the opposite. We get. Thus, the set of all solutions to this inequality is the interval.

Example 28. Solve inequality ( by introducing a new variable).

Solution: Let . Then this inequality will take the form: or , whose solution is the interval .

From here. Since the function increases, then .

Didactic material.

Specify the set of solutions to the inequality:

1. ; 2. ; 3. ;

6. At what values x Do the points on the function graph lie below the straight line?

7. At what values x Do the points on the graph of the function lie at least as low as the straight line?

Solve the inequality:

8. ; 9. ; 10. ;

13. Specify the largest integer solution to the inequality .

14. Find the product of the largest integer and the smallest integer solutions to the inequality .

Solve the inequality:

15. ; 16. ; 17. ;

18. ; 19. ; 20. ;

21. ; 22. ; 23. ;

24. ; 25. ; 26. .

Find the domain of the function:

27. ; 28. .

29. Find the set of argument values ​​for which the values ​​of each of the functions are greater than 3:

And .

Answers: 11.3; 12.3; 13. -3; 14.1; 15. (0; 0.5); 16. ; 17. (-1; 0)U(3; 4); 18. [-2; 2]; 19. (0; +∞); 20. (0; 1); 21. (3; +∞); 22. (-∞; 0)U(0.5; +∞); 23. (0; 1); 24. (-1; 1); 25. (0; 2]; 26. (3; 3.5)U (4; +∞); 27. (-∞; 3)U(5); 28. )