Graphical solution of exponential equations and inequalities. Solving exponential inequalities: basic methods

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 the exponential function, a technique for solving the simplest exponential equations

Let us recall the definition and basic properties of the 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 of the degree and Here x is an independent variable, an argument; y - dependent variable, function.

Rice. 1. Exponential function graph

The graph shows increasing and decreasing exponents, illustrating the exponential function when the base is greater than one and less than one, but greater than zero, respectively.

Both curves pass through the point (0; 1)

Exponential function properties:

Domain: ;

Range of values:;

The function is monotonic, as it increases, as it decreases.

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

When the argument increases from minus to plus infinity, the function increases from zero not 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. Solution of typical exponential equations

Let us 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.

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 considering 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 the simplest, variable changes are often used.

Let's use the property of degree:

We introduce a replacement. Let, then

We multiply the resulting equation by two and transfer 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 bring the degrees to the same indicator:

We introduce a replacement:

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

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

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

We get:

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

Let us examine 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, but it is 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 we get:

Change of variables should be introduced, we get a quadratic equation for y:

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

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:

Determine the roots by Vieta's theorem:

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

We will use the properties of the degree and reduce all degrees to simple bases:

It is easy to see the functions f and g:

Methods for solving systems of equations

To begin with, let's briefly recall what methods of solving systems of equations exist in general.

Exists four main ways solutions of systems of equations:

Substitution method: any of these equations is taken and $ y $ is expressed through $ 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 it is necessary to multiply one or both equations by such numbers that when both are added together, one of the variables “disappears”.

Graphical method: both equations of the system are displayed on the coordinate plane and the point of their intersection is found.

Method of introducing new variables: in this method, we replace any 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 by examples.

Example 1

Solve system of equations

Picture 1.

Solution.

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

Figure 2.

Substitute $ y $ in the second equation:

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

Answer: $(-4,6)$.

Example 2

Solve system of equations

Figure 3.

Solution.

This system is equivalent to the system

Figure 4.

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

Figure 5.

Let's 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 got 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 by 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 theorem on the equivalence of exponential inequalities:

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

\ U \ \

Answer: $(-4,6)$.

Example 2

Solve system of equations

Figure 3.

Solution.

This system is equivalent to the system

Figure 4.

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

Figure 5.

Let's 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 got 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 by 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 theorem on the equivalence of exponential inequalities:

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

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